Chip-firing game and a partial Tutte polynomial for Eulerian digraphs

Chip-firing game and a partial Tutte polynomial for Eulerian digraph Kévin Perrot Aix Mareille Univerité, CNRS, LIF UMR 7279 3288 Mareille cedex 9, France. kevin.perrot@lif.univ-mr.fr Trung Van Pham Intitut für Computerprachen, Theory and Logic Group Techniche Univerität Wien, Autria. Department of Mathematic for Computer Science Intitute of Mathematic, VAST, Vietnam pvtrung@logic.at, pvtrung@math.ac.vn Submitted: Dec 2, 203; Accepted: Mar 8, 206; Publihed: Mar 8, 206 Mathematic Subject Claification: 05C20, 05C3, 05C45, 05C57, 9A46 Abtract The Chip-firing game i a dicrete dynamical ytem played on a graph, in which chip move along edge according to a imple local rule. Propertie of the underlying graph are of coure ueful to the undertanding of the game, but ince a conjecture of Bigg that wa proved by Merino López, we alo know that the tudy of the Chip-firing game can give inight on the graph. In particular, a trong relation between the partial Tutte polynomial T G (, y) and the et of recurrent configuration of a Chip-firing game (with a ditinguihed ink vertex) ha been etablihed for undirected graph. A direct conequence i that the generating function of the et of recurrent configuration i independent of the choice of the ink for the game, a it characterize the underlying graph itelf. In thi paper we prove that thi property alo hold for Eulerian directed graph (digraph), a cla on the way from undirected graph to general digraph. It turn out from thi property that the generating function of the et of recurrent configuration of an Eulerian digraph i a natural and convincing candidate for generalizing the partial Tutte polynomial The author ha received funding from FONDECYT Potdoctorado 204 grant number 340527. The author ha received funding from the project P27600 of the Autrian Science Fund (FWF), the European Reearch Council under the European Community Seventh Framework Programme (FP7/2007-203 Grant Agreement No. 257039) and the Vietnamee National Foundation for Science and Technology Development (NAFOSTED). the electronic journal of combinatoric 23() (206), #P.57

T G (, y) to thi cla. Our work alo give ome promiing direction of looking for a generalization of the Tutte polynomial to general digraph. Keyword. Chip-firing game, critical configuration, Eulerian digraph, feedback arc et, recurrent configuration, reliability polynomial, Sandpile model, Tutte polynomial. Introduction The Tutte polynomial i an important polynomial in two variable which i defined on undirected graph [26]. It ha many intereting propertie and application. It evaluation at ome pecific point count variou combinatorial object uch a vertex coloring, panning tree, panning ubgraph, acyclic orientation, etc [5, 2, 4, 5, 7, 25]. There i an evident interet in looking for analogue of the Tutte polynomial for directed graph (digraph) and ome other object [7,, 3]. Thee attempt hare propertie of the Tutte polynomial. Neverthele, they are not natural extenion of the Tutte polynomial in the ene that one doe not know a converion from the propertie of thee polynomial to thoe of the Tutte polynomial, in particular how to get back to the Tutte polynomial on undirected graph from thee polynomial. For thi reaon the author of [7] aked for a natural generalization of Tutte polynomial for digraph. The evaluation of the Tutte polynomial T G (x, y) at x = i important ince it ha a trong connection to the reliability polynomial that i tudied in network theory. In thi paper we preent a polynomial that can be conidered a a natural generalization of T G (, y) for the cla of Eulerian digraph. An Eulerian digraph i a trongly connected digraph in which each vertex ha equal in-degree and out-degree, or equivalently there i a cloed path covering all arc. A connected undirected graph can be regarded a an Eulerian digraph by replacing each edge e by two revere arc a and b which have the ame endpoint a e. When conidering undirected graph een a Eulerian digraph in that way, we will ee that we get back to the partial Tutte polynomial T G (, y), which i a new and relevant feature. Thi work i baed on an idea conjectured by Bigg and proved by Merino López, that the generating function of the et of recurrent configuration of the Chip-firing game of an undirected graph i equal to the partial Tutte polynomial T G (, y) [, 9]. Baed on a dicrete dynamical ytem, thi contruction define a polynomial that characterize the graph upporting the dynamic. It i not traightforward to generalize thoe idea to the cla of Eulerian digraph, but the reult we will develop give a promiing direction for further extenion. The Chip-firing game i a dicrete dynamical ytem which i defined on a directed graph (digraph) G. The game i initialized by ome chip tored on the vertice of G. If a vertex ha at leat one out-going arc and ha a many chip a it out-degree, it can ditribute it chip to it neighbor by ending one chip along each out-going arc to the correponding vertex. The game playing with thi rule i called Chip-firing game (CFG). A ditribution of chip on vertice i called a configuration of G. Stabilizing a the electronic journal of combinatoric 23() (206), #P.57 2

configuration i the proce of repeatedly applying the rule whenever it i applicable. Thi proce may be infinite. If the proce i finite, we call the game convergent. It i known that the game either play forever or converge to a unique table configuration, in which the rule i no longer applicable [3, 4, 6]. The unique table configuration (if it exit) i called tabilization of the game. In thi paper we are intereted in the digraph with a global ink, i.e. a vertex of out-degree 0 and for any other vertex v there i a path from v to. The game on uch digraph i alway convergent for any initial configuration. For a trongly connected digraph (without global ink) the game play forever if the number of chip in the initial configuration i ufficient, but we can make the game alway convergent by chooing a vertex and removing all out-going arc of. The vertex become a global ink of the new graph. The game now i played on that new graph, and therefore it i alway convergent. In thi paper we tudy propertie of uch game, which are independent of the choice of, and provide clue to define a natural analogue of the Tutte polynomial, for the cla of Eulerian digraph. The Dollar game i a variant of CFG on undirected graph in which a particular vertex i called ink, and thi vertex can only be fired if all other vertice are table (not firable) [2]. In thi model the number of chip tored in the ink may be negative. Thi definition lead naturally to the notion of recurrent configuration (originally called critical configuration) that are table, and unchanged under firing at the ink and taking the tabilization of the reulting configuration. The Dollar game can be defined for Eulerian digraph with the ame definition, i.e. ome vertex i choen to be the ink that can only be fired if all other vertice are table [6]. Although the notion of recurrent configuration i originally defined on the Dollar game, it can be defined in an equivalent way on the Chip-firing game. Since we work only with the Chip-firing game throughout thi paper, we will ue the Chip-firing game to define recurrent configuration. Thi important notion ha a generalization to the digraph with a global ink [6]. The et of recurrent configuration of a CFG with a ink on an undirected graph ha many intereting propertie, uch a it i an Abelian group with the addition defined by vertex-wie addition of chip content followed by tabilization, and it cardinality i equal to the number of panning tree of the upport graph, etc. A direct conequence of the reult of Merino López for proving Bigg conjecture i that the generating function of recurrent configuration of a CFG with a ink i independent of the choen ink, and thu characterize the upport graph. Thi fact i definitely not trivial, and opened a new direction for tudying graph uing the Chip-firing game a a tool [8, 20]. A lot of propertie of recurrent configuration on undirected graph can be extended to Eulerian digraph without difficulty. However the ituation i different when one trie to extend the ink-independence property of the generating function to a larger cla of graph, in particular to Eulerian digraph, mainly becaue a natural definition of the Tutte polynomial for digraph i unknown, even for Eulerian digraph. In thi paper we develop a combinatorial approach, baed on a level-preerving bijection between two et of recurrent configuration with repect to two different ink, to how that thi ink-independence property alo hold for Eulerian digraph. Thi bijection provide new inight into the group of recurrent configuration. the electronic journal of combinatoric 23() (206), #P.57 3

It turn out from the ink-independence property of the generating function, that thi latter i a characteritic of the upport Eulerian digraph, and we can denote it by T G (y) regardle of the ink. We will ee that evaluation of T G (y) can be conidered a extenion of T G (, y) to Eulerian digraph, which make u believe that the polynomial T G (y) i a natural generalization of T G (, y). Furthermore, the mot important feature i that T G (y) and T G (, y) are equal on undirected graph. It require to be inventive to dicover which object the evaluation of T G (y) count, and we hope that further propertie will be found. The cla of Eulerian digraph i in-between undirected and directed graph, and following the track we develop in thi paper, we propoe ome conjecture that would be promiing direction of looking for a natural generalization of T G (x, y) to general digraph. The paper i divided into the following ection. Section 2 recall the definition of the Chip-firing game on digraph and ome known reult about the recurrent configuration on a digraph with global ink and on an Eulerian digraph with a ink. Section 3 etablihe the ink-independence of the generating function of recurrent configuration in the cae of Eulerian digraph. The Tutte polynomial generalization i preented in Section 4, and Section 5 hint at continuation of the preent work. 2 Chip-firing game and recurrent configuration All graph in thi paper are aumed to be multi-digraph without loop. Throughout thi paper an undirected graph i regarded a a digraph by replacing each edge e by two revere arc that have the ame endpoint a e. Graph with loop will be conidered in Section 4. In thi ection we recall the definition of the Chip-firing game and preent ome known reult about recurrent configuration of a CFG on a graph with a global ink and on an Eulerian graph with a ink, followed by traightforward conideration on the number of chip tored on vertice of recurrent configuration. All graph in thi ection are aumed to be connected. 2. Chip-firing game For a digraph G = (V, A) and an arc e A, we denote by e and e + the tail and head of e, repectively. For a vertex v let deg + G (v) denote the number of arc e with e = v and deg G (v) denote the number of arc e with e+ = v. For two vertice v, w V, let deg G (v, w) denote the number of arc from v to w in G. A configuration c on G i a map from V to N = {0,, 2,... }. A vertex v i firable in c if and only if c(v) deg + G (v) > 0. A configuration c i called table if there i no firable vertex in c. Firing a firable vertex v i the proce that decreae c(v) by deg + G (v) and increae each c(w) with w v by deg G (v, w). A equence (v, v 2,, v k ) of vertice of G i called a firing equence of a configuration c if tarting from c we can conecutively fire the vertice v, v 2,, v k in thi order. Applying the firing equence lead to configuration d and we write c v,v 2,,v k d, or c d without pecifying the firing equence. A ink i a vertex of out-degree 0. A vertex i called global ink if it i a ink and from every other vertex there i a directed path leading to. Note that if G ha a global ink then the ink i unique. the electronic journal of combinatoric 23() (206), #P.57 4

Lemma. [3, 4, 6] Suppoe that G ha a global ink. For any initial configuration c the game converge to a unique table configuration, denoted by c. Let f and g be two firing equence of c uch that c f c and c g c. Then for any vertex v the number of time v occur in f i the ame a in g. The configuration c in the lemma i called the tabilization of c. 2.2 Recurrent configuration on a digraph with a global ink The following i imple but very important, and will often be ued without explicit reference. Lemma 2. Let G = (V, A) be a graph with a global ink. For two configuration c and d, we denote by c + d the configuration given by (c + d)(v) = c(v) + d(v) for any v V. Then c + d = c + d. For two et X and Y we denote by Y X the et of map from X to Y. In the ret of thi ubection we work with a graph G = (V, A) which ha a global ink. The definition of recurrent configuration i baed on the convergence of the game, which i enured if G ha a global ink. Since i not firable no matter how many chip it ha, it make ene to define a configuration to be an element in N V \{}. When a chip goe into the ink, it vanihe. Two configuration are conidered to be the ame if they have the ame number of chip on every vertex except for the ink. Note that in thi ection we conider only one fixed ink, but in ubequent ection we will conider the CFG relative to different choice of ink, and therefore we will need ome more notation. Let u not be overburdened yet, a configuration on G with ink i an element in N V \{}. Definition. A table configuration c i recurrent if and only if for any configuration a there i a configuration b uch that c = a + b. There are everal equivalent definition of recurrent configuration. For an undirected graph G (regarded a a digraph) with a particular vertex, which i called a ink, let H be the graph which i obtained from G by removing all out-going arc of. Then recurrent configuration of H can be defined by the firing rule of the Dollar game on G a follow. A configuration i recurrent if it remain unchanged under firing the ink and tabilizing the reulting configuration [2, 0]. For a graph G with global ink, recurrent configuration can be defined naturally by uing the notion of recurrent tate of the Markov chain which i defined a follow. The tate pace i the et of all table configuration of G. When at a tate c, the next tate i obtained by adding one chip to c at ome vertex (ditinct from the ink) choen uniformly at random, and then tabilizing the reulting configuration. The recurrent configuration can be defined to be the recurrent tate of thi Markov chain [9, 0, 8]. The one we preent in thi paper ay that c i recurrent if and only if it can be reached from any other configuration a by adding ome chip (according to b) and then tabilize the reulting configuration. Dhar proved that the et of recurrent configuration ha an elegant algebraic tructure [9]. Fix a linear order v v 2 v n v n on the vertice, where n = V. We the electronic journal of combinatoric 23() (206), #P.57 5

uppoe that v n =. Now a configuration of G can be repreented a a vector in Z n. The Laplacian matrix of G i an n n matrix which i defined by { deg ij = G (v i, v j,) if i j deg + G (v i) if i = j. Note that the Laplacian matrix of a general graph (not necearily having a global ink) i defined by the ame formula a above. Let Λ be the matrix which i obtained from by deleting the lat row and column of. Let Λ i denote the ith row of Λ. Firing the index i in a configuration c correpond to adding the vector Λ i to c. We define a binary relation over Z n by b c if and only if there exit a, a 2,, a n Z uch that b c = i n a iλ i, i.e. b and c are linked by a (poibly impoible to perform) equence of firing. The following reult tate the nice algebraic tructure of the et of all recurrent configuration of G with ink. Lemma 3. [6] The et of all recurrent configuration of G i an Abelian group under the operation (a, b) a + b. Thi group i iomorphic to Z n / Λ, Λ 2,, Λ n. Moreover, each equivalence cla of Z n / contain exactly one recurrent configuration, and the number of recurrent configuration i equal to the number of equivalence clae. The group in Lemma 3 i called the Sandpile group of G. For two function f, g : X R we write f g if f(x) g(x) for every x X. The following imple propertie can be derived eaily from the definition of recurrent configuration. Lemma 4. The following hold:. Let c be a configuration uch that c(v) deg + G (v) for every v. Then c i recurrent. 2. Let c and d be two configuration (element of N V \{} ) uch that c d. Then v c(v) v c(v) v d(v) d(v). v Proof. Moreover, if c i recurrent then d i alo recurrent.. For any configuration a let b = c a. Clearly, b i a configuration. We have a + b = a + c a = a + c a = c, therefore c i recurrent. 2. Let f = (v, v 2,, v k ) be a firing equence of c uch that c f c. Since v c(v) c(v) i the number of chip lot into the ink, we have v v c(v) v c(v) = i k deg G (v i, ). the electronic journal of combinatoric 23() (206), #P.57 6

Since c(v) d(v) for any v, f i alo a firing equence of d. Therefore there i a firing equence g = (v, v 2,, v k, v k+, v k+2,, v l ) of d uch that d g d. For the ame reaon we have v d(v) d(v) = deg G (v i, ). v i l The firt claim follow. Let a be an arbitrary configuration. Since c i recurrent, there i a configuration b uch that a + b = c. Let e = b + d c. We have a + e = a + b + d c = a + b + d c = c + d c = c + d c = d, thu d i recurrent. 2.3 Recurrent configuration of an Eulerian digraph with a ink In thi ubection we work with an Eulerian graph G and preent propertie that recurrent configuration have in that cae. A in the previou ubection, the definition of recurrent configuration i baed on the convergence of the game. Therefore a global ink play an important role in the definition. The digraph G i trongly connected, therefore it ha no global ink and the game may play forever from ome initial configuration. To overcome thi iue, we ditinguih a particular vertex of G that play the role of the ink. By removing all outgoing arc of from G we obtain the digraph H that ha a global ink. The Chip-firing game on G with ink i the ordinary Chip-firing game that i defined on H, and recurrent configuration are defined a preented in the previou ubection, on H. Figure a and b preent an example of G and H. It i a good way to think of the Chip-firing game on an Eulerian digraph with a ink a the ordinary Chip-firing game on G with a fixed vertex that never fire in the game no matter how many chip it ha. In thi ection we will work with G, and H. A configuration of the Chip-firing game on G with ink i a map in N V \{}. To verify the recurrence of a configuration c, we have to tet the condition that for any configuration a there i a configuration b uch that a + b = c. Thi i a tireome tak. However, in the cae of Eulerian digraph we have the following ueful criterion. Lemma 5. [9, 6] A configuration c i recurrent if and only if c + β = c, where β i the configuration defined by β(v) = deg G (, v) for every v. Moreover, if c i recurrent then each vertex ditinct from occur exactly once in any firing equence f from c + β to c + β. Figure c preent a configuration c. The configuration c + β i preented in Figure d, adding β correpond to firing the ink. To verify the recurrence of c, one compute c + β. Starting with c + β we fire conecutively the vertice v, v 3, v 2, v 4 in thi order and get exactly the configuration c, therefore c i recurrent. Thi procedure i called Burning algorithm. For two integer p, q we denote by [p..q] the et {x Z : p x q}. The following lemma will be important later. the electronic journal of combinatoric 23() (206), #P.57 7

t u t u v r v r (a) G (b) H 0 0 2 0 2 (c) A configuration c (d) c + β Figure : Burning algorithm Lemma 6. Let c and d be two table configuration uch that c and d are in the ame equivalence cla. If c i recurrent then d(v) v v c(v). Proof. Let β be the configuration that i defined a in Lemma 5 and a be an arbitrary table configuration. We claim that for any firing equence f = (v, v 2,, v k ) of a + β uch that a + β f a + β, each vertex of G occur at mot once in f. For a contradiction we aume otherwie. Thi aumption implie that there i a firt repetition, i.e., there i p [..k] uch that v, v 2,, v p are pairwie-ditinct and v p = v q for ome q [..p ]. We denote b the configuration obtained from a + β after the vertice v, v 2,, v p have been fired. We will now how that v p i not firable in b, a contradiction. Let r be the number of chip v p ha received from it in-neighbor when the vertice v, v 2,, v p have been fired. Since adding β correpond to firing the ink and v p ha been fired exactly once during the proce of firing vertice in the equence (v, v 2,..., v p ), it follow that b(v p ) = a(v p ) + deg G (, v p ) + r deg + G (v p). Since v, v 2,, v p are pairwie-ditinct and different from the ink, we have r + deg G (, v p ) deg G (v p). The digraph G i Eulerian, therefore deg G (v p) = deg + G (v p) and from the previou equality we have b(v p ) a(v p ), but a i table o vertex v p i not firable in configuration b, which i aburd. Since each of the in-neighbor of i fired at mot once in any firing equence f of a + β uch that a + β f a + β, it follow that during the tabilization proce of a + β the number of chip lot in the ink i not greater than deg G () = deg+ G (). Thi implie that + β)(v) v (a a + β(v) deg + G (). v the electronic journal of combinatoric 23() (206), #P.57 8

0 0 0 2 (a) A recurrent configuration (b) A non-recurrent configuration Figure 2: Two table configuration from the ame equivalence cla on an undirected graph Since it follow that v (a + β)(v) = v a(v) + v v a(v) v β(v) = v a + β(v). Repeating the application of thi inequality n time we have v a(v) a + nβ(v), v a(v) + deg + G () where nβ i the configuration given by (nβ)(v) = n β(v) for any v. Thi reaoning can be applied to d and we have v d(v) d + nβ(v) for any n N. v Since for any vertex v and any w being an out-neighbor of there i a path in H from w to v, with a ufficiently large n there i an appropriate firing equence g of d + nβ and a configuration e uch that d + nβ g e and e(v) deg + G (v) for any v. When tabilizing e, it follow from Lemma (convergence), Lemma 4 (recurrence) and Lemma 3 (uniquene of recurrent configuration in an equivalent cla) that it lead to c, that i, c = e = d + nβ. Thi finihe the proof. Quetion. Doe the claim of Lemma 6 hold for a general digraph with a global ink? Note that if thi tatement i true, then it i tight. Figure 2 preent an example, on an undirected graph, of a recurrent configuration and a non-recurrent configuration belonging to the ame equivalence cla, uch that they contain the ame total number of chip. A a conequence, the recurrent configuration i not necearily the unique configuration of maximum total number of chip over table configuration of it equivalence cla. the electronic journal of combinatoric 23() (206), #P.57 9

0 0 0 0 0 0 0 0 0 0 0 0 0 Figure 3: Recurrent configuration with repect to ink 0 0 0 0 0 0 2 0 0 2 0 2 0 Figure 4: Recurrent configuration with repect to ink r 3 Sink-independence of generating function of recurrent configuration of an Eulerian digraph Key obervation. Let u give an important obervation that motivate the tudy preented in thi paper. We conider the Chip-firing game on the digraph drawn on Figure a. In thi game the vertex i choen to be ink. All the recurrent configuration are preented in Figure 3. For each recurrent configuration we compute the um of chip on the vertice different from the ink. We get the orted equence of number (,, 2, 2, 2, 3). If r i choen to be the ink of the game, all the recurrent configuration are given in Figure 4, and the um of chip on vertice different from the ink give the orted equence (2, 2, 3, 3, 3, 4). The two equence are the ame up to adding a contant equence. Thi property alo hold with other choice of ink, therefore, up to a contant, thi equence i characteritic of the upport graph itelf. Thi intereting property i the main dicovery exploited in thi paper, and allow to generalize the contruction preented in [] and proved in [9] of an analogue for the Tutte polynomial to the cla of Eulerian digraph. It i tated in the following theorem. the electronic journal of combinatoric 23() (206), #P.57 0

Theorem. Let G be an Eulerian digraph and a vertex of G. For each recurrent configuration with repect to ink, let um G (c) denote deg + G ()+ v c(v). Let c, c 2,..., c p be an enumeration of recurrent configuration with ink. Then the equence (um G (c i )) i p i independent of the choice of up to a permutation of the entrie. It make ene to call an Eulerian digraph G undirected if for any two vertice v, w of G we have deg G (v, w) = deg G (w, v). The reult of Merino López [9] implie that Theorem i true for undirected graph. The following known reult i thu a particular cae of Theorem, for the cla of undirected graph. Theorem 2. [9] Let C be the et of all recurrent configuration with repect to ome ink. If G i an undirected graph (defined a a digraph) then T G (, y) = c C y level(c), where T G (x, y) i the Tutte polynomial of G and for any c C, level(c) = A 2 + deg+ G () + c(v). v In the ret of thi ection we work with an Eulerian digraph G = (V, A). For implicity we remove G in ome notation uch a deg + G (v), deg G (v), um G(c), deg G (v, w). In order to prove Theorem, we conider the following natural approach. Let r and be two arbitrary ditinct vertice of G. Let U = V \{r} and W = V \{}. We denote by R and S the et of all recurrent configuration with repect to ink r and, repectively. We are going to contruct a bijection θ from R to S uch that um(c) = um(θ(c)) for every c R. Clearly, thi bijection will imply Theorem. We recall that it follow from [6] that R = S. For each configuration c N V we denote by c r (rep. c ) the tabilization of c with repect to ink r (rep. ink ). Note that in the proce of tabilizing c with repect to ink r, when a chip arrive at r it doe not vanih but tay at r forever ince r i never fired in the proce. See Figure 5 for an illutration. Warning: let u underline that a configuration i now an element of N V. Notation. For a function f : X Y and a ubet Z X we denote by f Z the retriction of f to Z. We denote by ˆR the et of configuration c N V uch that c U R and c(r) deg + (r), and by Ŝ the et of configuration c NV uch that c W S and c() deg + (). For each c R (rep. S) let c denote the configuration in ˆR (rep. Ŝ) uch that c U = c (rep. c W = c) and c(r) = deg + (r) (rep. c() = deg + ()). The following how the firt relation between ˆR and Ŝ under the tabilization. Lemma 7. Let c R, then c Ŝ. Symmetrically, for each c S we have cr ˆR. the electronic journal of combinatoric 23() (206), #P.57

(a) A configuration c of G (b) c Figure 5: Chip-unvanihed tabilization 0 2 0 0 2 0 (a) A recurrent configuration c 0 4 2 (b) Configuration c 0 0 0 0 0 (c) c (d) Recurrent configuration c W Figure 6: Change of ink Since the concept of thi lemma i at the heart of the contruction of the map θ, we give an illutration of the claim before going into the detail of the proof. We conider the Eulerian digraph given in Figure. Figure 6a how a recurrent configuration with repect to ink r. The configuration c i given in Figure 6b. Figure 6c and Figure 6d how c and it retriction to W. Uing the Burning algorithm, one eaily check that the configuration in Figure 6d i indeed recurrent with repect to ink. Proof of Lemma 7. We will once again ue Lemma 5 (the Burning algorithm), which provide a firing equence aociated to a recurrent configuration of R, that we will manipulate to built a firing equence aociated to a recurrent configuration of S. In c, only r i firable, and after firing it, we will ue the firing equence leading back to c, provided by Lemma 5. Let β N U (rep. γ N W ) be given by β(v) (rep. γ(v)) i equal to deg(r, v) (rep. deg(, v)). Let a be uch that c r a. We have a U = c + β, and can therefore apply Lemma 5 (ince c i recurrent), providing a firing equence f = (v, v 2,, v V ) of c uch that v = r and each vertex of G occur exactly once in thi equence. Let k be uch the electronic journal of combinatoric 23() (206), #P.57 2

that = v k and b be the configuration reached after vertice (v, v 2,..., v k ) have been fired. Vertex i firable in b, thu b() deg + (). Since the game i convergent, and thi intermediate configuration b i reachable from c without firing, when we tabilize c with repect to ink we end up with at leat a many chip in a in b, and therefore c () deg + (). Since c f c, the equence g = (, v k+, v k+2,, v V, v, v 2,, v k ) i a firing equence of b. We now conider the game with repect to ink (that i, on G where the out-going arc of are removed), and the configuration b W. Let d be uch that b d. We have d W = b W + γ, and the ret of the firing equence implie that b W + γ b W, therefore b W + nγ b W for any n N. The recurrence of b W can be hown imilarly a in the proof of Lemma 6 a follow. With n large enough there i e N W uch that b W e and e(v) deg + (v) for any v W. By Lemma 4 we have e = b W i recurrent. Since c () deg + () and b W = c W, it follow that c Ŝ. Lemma 7 naturally ugget a bijection from R to S that i defined by c c W. However, thi doe not give the intended bijection ince it doe not necearily preerve the um of chip, a hown on Figure 6. We will improve the above map by adding ome extra chip to c at the vertex r o that the map preerve the um. The required number of extra chip added to r follow from Lemma 0 below and i a non-contructive quantity. That i what we are going to preent now. We need the following notation. We denote by r (rep. ) the configuration on G which ha one chip at r (rep. ) and none in other vertice. By thi notation for any i N we have i r (rep. i ) i the configuration which ha i chip at r (rep. i chip at ) and none in other vertice. Lemma 8. For any c ˆR we have c Ŝ. Symmetrically, for any c Ŝ we have cr ˆR. Proof. Clearly, we have c U c. It follow from Lemma 7 and the econd item of Lemma 4 that c W S. By uing the ame argument a in the proof of Lemma 7 we have c () deg + (). Thi finihe the proof. The lemma mean that the correpondence c c i a map from ˆR to Ŝ. The following implie the injectivity of thi map. Lemma 9. For any c ˆR we have (c ) r = c. Proof. Let a denote (c ) r. It follow from Lemma 8 that both a and c are in ˆR. Since G i Eulerian and a can be obtained from c by a equence of firing, it follow that a U and c U are in the ame equivalence cla. Both are recurrent, hence from Lemma 3 they are equal. Since a and c contain the ame total number of chip, and are equal on the vertice different from r, it follow that a = c. The aim i now to find, for every recurrent configuration c R, the good i o a to get a bijection from R to S that preerve the um of chip. We firt concentrate on the um conervation: if one want to have um(c) = um ( c + i r W ), the electronic journal of combinatoric 23() (206), #P.57 3

then the number i mut be choen o that c + i r () = i + deg + (), becaue the i extra chip are not counted in both um in thi cae. The following how that uch an i alway exit. Lemma 0. For every c R there exit i uch that c + i r () = i + deg + (). Proof. For each i N let d i denote the c + i r. Let the function f : N Z be defined by f(i) = d i () deg + () i. We are going to prove that there exit j N uch that f(j) = 0. Since c + i r c + (i + ) r, it follow from Lemma 4 (uing the trick v W c(v) = v V c(v) c()) that d i() c() d i+ () c(), therefore d i () d i+ (). A a conequence f(i + ) f(i) for every i, that i, the function f decreae by at mot one. By Lemma 8 we have d 0 () deg + (), therefore f(0) 0. Since f(i + ) f(i) for any i N, the proof i completed by howing that there i j N uch that f(j) 0. In particular, we are going to prove that f(n ) 0, where N = S. Note that N i the order of the Sandpile group of G with repect to ink. f(n ) = d N () deg + () (N ). We are going to ue Lemma 6, which tate that the recurrent configuration ha maximum total number of chip over table configuration of it equivalence cla, in order to upper bound d N () by c() + N, and the reult follow ince vertex i table in the recurrent configuration c (meaning that c() deg + () ). Let a denote c + (N ) r and b denote c r. Note that b N V ince G i an Eulerian graph. We have a = b+n r, thu the choice of N implie that a and b are in the ame equivalence cla with repect to ink, and the firt contain N more chip than the latter. Since d N = a and the configuration d N W i recurrent, it follow from Lemma 6 that v W b(v) v W d N (v). It remain to exploit the total number of chip difference between the two configuration: N + v V b(v) = v V d N (v). Replacing v W x(v) by v V x(v) x() on both ide, the inequality given by Lemma 6 thu become b() + N d N (), and equivalently c() + N d N (). We can now contruct the intended bijection θ. For each c R (rep. c S), let I(c) (rep. J (c)) denote the mallet number i N uch that c + i r () = deg + () + i (rep. c + i r (r) = deg + (r) + i). The poitive integer I(c) and J (c) are called the wap number of c from r to and from to r, repectively. By Lemma 0 we know that wap number are well-defined and unique. The map θ atifie θ : R S c d W, where d = c + I(c) r. um(c) = um(θ(c)). Since R = S i finite, in order to prove Theorem it remain to how that θ i injective. Let u firt preent ome propertie of wap number. A configuration c R i called minimal if there i no configuration d R uch that c d and d c, and minimum if v U c(v) i minimum over all configuration in R. the electronic journal of combinatoric 23() (206), #P.57 4

Propoition. Let c R. If c i minimum then I(c) = 0. Proof. Let a denote c. By definition of I(c), the aim i to prove that a() = deg + (). The proof relie intuitively on Lemma 7: it ay that the lower bound for a() i alway reached for the configuration containing the minimum total number of chip. Let u aume it i fale, therefore we can define b a uch that b(v) = a(v) for v W and b() = deg + (), and we will get a contraction to the minimality of c. By Lemma 8 we have a() deg + (). By the aumption b a we have a() > deg + (). Thi implie that b(v) < a(v) = c(v). v V v V v V By Lemma 8 the configuration a i in Ŝ, and o i b. Applying again Lemma 8 to and S, we have b r ˆR, and therefore b r (r) deg + (r) = c(r). Now, ince c(v), it follow that v V b r (v) = v V b(v) < v V b r (v) < c(v) = c(v), v U v U v U a contradiction to the minimality of c. Propoition 2. Let c, d R. If c d then I(c) I(d). Proof. Regarding Propoition, we would intuitively expect that the number I(x) increae monotonically with the total number of chip of x. We ue the ame contruction a in the proof of Lemma 0. Let k denote I(d). Let a and b denote c + k r and d + k r, repectively. Since c + k r d + k r, it follow from Lemma 4 that a() c() b() d(). Since by hypothei c() d(), we have a() b(), therefore a() deg + () k b() deg + () k = 0. Let f : N Z be given by f(i) = ( c + i r ) () deg + () i, from the previou inequality it verifie that f(k) 0, and I(c) i by definition the mallet j uch that f(j) = 0. By the ame argument a in the proof of Lemma 0, we have f(0) 0 and f(i + ) f(i) for any i N. A a conequence, there i j [0..k] uch that f(j) = 0, therefore I(c) k = I(d). By Propoition and 2, for a recurrent configuration c R the number I(c) increae monotonically a we add chip to c, tarting from 0 when the configuration i minimum. When G i an undirected graph, every minimal recurrent configuration i minimum ince every minimal recurrent configuration ha the ame number of chip, namely A 2 deg+ () [23]. Thu I(c) = 0 for every minimal recurrent configuration c. One tend to think that a minimal configuration c hould alo have I(c) = 0, but it may indeed be trictly poitive a hown on Figure 7. the electronic journal of combinatoric 23() (206), #P.57 5

r 0 (a) An Eulerian graph (b) A minimal recurrent configuration with repect to ink r Figure 7: A minimal recurrent configuration c with I(c) = Quetion. Give an upper bound of I(c) when c i minimal. There i a nice relation between wap number for c in R from r to, and for θ(c) from to r. The following propoition doe mot part of the work to prove Theorem, and the latter can be conidered a a corollary of thi reult. Propoition 3. For all c R, we have I(c) = J (θ(c)). Proof. The propoition i proved with two inequalitie. J (θ(c)) I(c): Let l denote I(c) and let a = c + l r. We have a W = θ(c). Firt, by definition of l we have a() = deg + () + l, therefore a = θ(c) + l. Second, by Lemma 9 applied to c + l r, we have a r = c + l r r = c + l r. Thi implie that ã W + l r(r) = a r (r) = deg + (r)+l. Since J (a W ) i the minimal number i uch that ã W + i r(r) = deg + (r)+i, it follow that J (θ(c)) = J (a W ) l. J (θ(c)) I(c): Thi part of the lemma i more involved. For convenience, let u denote θ = θ(c). Let p denote J (θ). The previou inequality implie p l. In order to get a contradiction, let u uppoe that p < l. Let b R be uch that θ + p r = b + p r (the exitence of b i due to Lemma 7 and the definition of p). The above inequality applied to θ implie that b c. We have c + l r = θ + l and b + p r = θ + p (By Lemma 9), () the electronic journal of combinatoric 23() (206), #P.57 6

therefore the two configuration c + l r W and b + p r W are in the ame equivalence cla for the CFG with ink. Removing p chip to r in both configuration doe not affect the equivalence relation, hence with k = l p > 0, c + k r W and b W are alo in the ame equivalence cla for the CFG with ink, and from Lemma 8 and the uniquene of the recurrent configuration in an equivalence cla (Lemma 3), c + k r W = b W. (2) From equation () there are k more chip in c + l r than in b + p r, thu it follow from the above equality (2) that c + k r () = b () + k. (3) We now conider the two configuration d and e defined by d = c + (k ) r d + r = c + k r and e = b r e + r = b. It follow from equality (2) that A we will ee, it i not poible that both: thee two configuration are equal; d + r W = e + r W. (4) enough chip go to during thee tabilization procee o that equation (3) i verified. Let u preent a reaoning contradicting equation (3). We firt work on the total chip content of d W and e W. For the ame reaon a above, d W and e W belong to the ame equivalence cla for the CFG with ink, and o do d W and e W becaue the firing proce doe not affect the equivalence relation. We have d W = c + (k ) r W with k 0, thu from Lemma 8 it i recurrent. Furthermore e W i table, and ince they belong to the ame equivalence cla, it follow from Lemma 6 that d (v) e (v). (5) v W v W the electronic journal of combinatoric 23() (206), #P.57 7

Now we compare the number of chip going into the ink. Let f = (v,..., v m ) and g = (w,..., w n ) be two firing equence uch that d + r f d + r and e + r g e + r. Obviouly / f and / g, and it follow from equation (4) and (5) that during the tabilization proce, more chip go to in f than in g: deg(v i, ) deg(w i, ). (6) i m i n In order to get the intended contradiction with (3), let u have a cloe look at the chip content in both ink, uing the fact that from the minimality of l, c + k r () = d + r () c + (k ) r () > deg + () + (k ). = d () + i m deg(v i, ) = c + (k ) r () + i m deg(v i, ) > deg + () + (k ) + i m deg(v i, ) equation (6) tability deg + () + (k ) + i n deg(w i, ) deg + () + (k ) + i n deg(w i, ) + e () deg + () + = k + e + r () = k + b (), which contradict equation (3). Theorem i now eay to prove. Proof of Theorem. Since R = S, it remain to prove that the map θ i injective. For a contradiction, uppoe it i not, that i, there exit c and d belonging to R and uch that c d and θ(c) = θ(d). By Propoition 3 we have I(c) = I(d). Let k denote I(c) and let e denote θ(c). It follow from Lemma 9 that a contradiction. c = c + k r r U = ẽ + k r U = d + k r r U = d, the electronic journal of combinatoric 23() (206), #P.57 8

4 Tutte-like propertie of generating function of recurrent configuration We preent in thi ection a natural generalization of the partial Tutte polynomial T G (, y) in one variable for the cla of Eulerian graph. In order to et up the mot general etting, we introduce it for the cla of Eulerian graph with loop. Note that loop are not intereting regarding the Chip-firing game: a loop imply freeze a chip on one vertex, that i the reaon why we did not conider them in previou ection. The Tutte-like polynomial we preent i contructed from the generating function of the et of recurrent configuration with repect to an arbitrary ink, and it uniquene i baed on the ink-independence property expoed in Theorem. Let u firt preent the extenion of Theorem to the cla of Eulerian graph with loop. We begin with the definition of the Chip-firing game for thi cla of graph. Note that the out-degree of a vertex v i the number of arc whoe tail i v, therefore include loop. Let G = (V, A) be an Eulerian graph poibly having loop. A vertex v i firable in a configuration c if c(v) deg + (v) and deg + (v) deg(v, v), where deg(v, v) i the number of loop at v. Firing a firable vertex v mean the proce that decreae c(v) by deg + (v) and increae each c(w) by deg(v, w) for all w, or equivalently decreae c(v) by deg + (v) deg(v, v) and increae each c(w) with w v by deg(v, w). The Burning algorithm preented in Lemma 5 remain valid for Eulerian graph with loop. A pointed out above, the CFG on a digraph poibly having loop i very cloe to the CFG on the digraph where the loop are removed. For a digraph G we denote by G the digraph G in which all loop are removed, and denote by L(G) the number of loop of G. Regarding undirected graph, the influence of a loop i the ame a a directed loop (it alo freeze one chip). For two arc a and b of G, a i revere of b if a = b + and a + = b. An undirected graph with loop i converted to an Eulerian graph in the ame way a an undirected graph without loop, however with the exception that each undirected loop e i replaced by exactly one directed loop that ha the ame endpoint a e. Theorem i generalized to the cla of Eulerian graph poibly having loop with the following lemma. From now on, we will alway conider an arbitrary fixed ink denoted, therefore a configuration mean an element in N V \{}. Lemma. Let G = (V, A) be an Eulerian graph with ink. Let C and C be the et of recurrent configuration of G and G with repect to ink, repectively. For each configuration c C, let µ(c) : V \{} N be given by µ(c)(v) = c(v) + deg(v, v) for any v. Then µ i a bijection from C to C. Moreover, c(v) µ(c)(v) = deg(v, v) for any c C. v v v Thi lemma can be proved eaily by uing the definition of recurrent configuration with the obervation that if a configuration c i recurrent with repect to G then c(v) deg(v, v) for any v. An illutration of Lemma i given in Figure 8. In the ret of thi ection, we work with an Eulerian graph G = (V, A) poibly having loop and an arbitrary but fixed vertex of G that play the role of ink for the game. We now introduce the partial Tutte polynomial generalization, which i defined a the the electronic journal of combinatoric 23() (206), #P.57 9

(a) G (b) G 2 3 0 0 (c) c C (d) µ(c) C Figure 8: Relation between C and C generating function of the et of recurrent configuration. The generating function i baed on the concept of level of recurrent configuration, which correpond to the previouly defined um normalized according to the mallet level of a recurrent configuration. For an Eulerian graph G, let κ(g) denote the minimum of um(c) = deg + () + v c(v) over all recurrent configuration c of G with repect to ink. Theorem implie that κ(g) i independent of the choice of. It follow from [23] that the problem of finding κ(g) i NP-hard for Eulerian graph, and a a conequence the Tutte-like polynomial we preent i alo NP-hard to compute. In addition, when G i undirected (and defined a a digraph i.e., each edge i repreented by two revere arc), the number κ(g) ha an exact formula, namely κ(g) = A(G). For a recurrent configuration c of G with repect to ink 2 we define level G (c) = um(c) κ(g). Thi i a generalization of the level that wa defined in [], becaue we recover the latter when G i undirected. Let C denote the et of all recurrent configuration of G with repect to ink. The generating function of C i given by T G (y) = c C y level G(c), and we claim that it i a natural generalization of the partial Tutte polynomial, for the cla of Eulerian graph. Firt, it follow from Theorem that T G (y) i independent of the the electronic journal of combinatoric 23() (206), #P.57 20

choice of, thu i characteritic of the upport graph G itelf. We are going to preent in thi ection a number of propertie of T G (y) that can be conidered a the generalization of thoe of the Tutte polynomial in one variable, namely T G (, y), that i defined on undirected graph. The mot intereting and new feature i that when G i an undirected graph we get back to the well-known Tutte polynomial. Thi fact i traightforward to notice. T G (y) = T G (, y) if G i an undirected graph. T G () count the number of oriented panning tree of G rooted at [6]. It generalize the evaluation T G (, ) that count the number of panning tree of an undirected graph. T G (0) count the number of maximum acyclic arc et with exactly one ink [23]. Therefore T G (0) i a natural generalization of T G (, 0) that count the number of acyclic orientation with a fixed ource of an undirected graph. For an undirected graph G, T G (, 2) count the number of panning connected ubgraph of G. It would be intereting to invetigate a combinatorial interpretation for T G (2). We propoe the following quetion for future work. Quetion. What doe T G (2) count? Thee evaluation et up a promiing ground for further invetigation, but it i definitely not trivial to find out the object counted by evaluation of graph polynomial. We are now going to preent the extenion to T G (y) of four known recurive formula for the Tutte polynomial in the undirected cae. We will need the two following imple lemma. For a ubet B of A let G\B denote the graph (V, A\B). We write G\e for G\{e} if e i an arc of G. For an arc e of G with two endpoint v and w let G/e denote the digraph that i made from G by removing e from G, replacing v and w by a new ingle vertex u, and for each remaining arc a if the head (rep. tail) of a in G i v or w then the head (rep. tail) of a in G/e i u. Thi procedure i called arc contraction. An analogue of arc contraction may alo be defined for vertice. For a ubet W of V, let G/W denote the digraph contructed from G by replacing all vertice in W by a new vertex w, and each arc e A uch that e W (rep. e + W ) in G by e = w (rep. e + = w) in G/W. In the cae of undirected graph Merino López gave a relation between the recurrent configuration of G and the recurrent configuration of the contracted graph G/e, where e i an edge of G [9]. In particular the author howed a bijection between the et of recurrent configuration which have maximum value at ome fixed neighbor t of the ink and the et of recurrent configuration of the contracted graph G/(, t) with the ink being the new vertex from the edge contraction. The following lemma i a generalization of that relation for the cla of Eulerian graph. The proof idea i originally due to Merino López [9]. We recall that C i the et of recurrent configuration of the CFG on G with repect to ink. the electronic journal of combinatoric 23() (206), #P.57 2

w a u e a v (a) G Figure 9: Arc contraction (b) G/e Lemma 2. Let W be a non-empty ubet of the et of out-neighbor of, i.e., for every v W we have v and (, v) A. Let t be the new vertex in G/W {} reulting from contracting the et of vertice W {}. For any c C, if c(v) deg + (v) deg(, v) for all v W, then c V \(W {}) i a recurrent configuration of G/W {} with repect to ink t. Converely, if d i a recurrent configuration of G/W {} with repect to ink t, then every configuration c : V \{} N, atifying c(v) = d(v) for all v V \(W {}) and deg + (v) > c(v) deg + (v) deg(, v) for all v W, i in C. Proof. Thi proof i traightforward. We ue Lemma 5 (Burning algorithm) and the hypothei that a configuration i recurrent, thu it admit a firing equence, in order to contruct a firing equence for the conidered configuration, which prove that it i recurrent (again by Lemma 5). We denote the vertice in W by w, w 2,, w q. Let the configuration β : V \{} N be given by β(v) = deg(, v) for any v V \{}. The condition c(w i ) deg + () deg(, w i ) for any i implie that w i i firable in c+β for any i. It follow from Lemma and Lemma 5 that there i a firing equence f = (v, v 2,, v k ) of c+β in G uch that c+β f c, v i for any i, each vertex of G ditinct from occur exactly once in f, and v i = w i for any i [..q]. Let a be uch that c + β w,w 2,,w p a. Let γ : V \(W {}) N be given by γ(v) = deg(t, v) for any v V \(W {}). Clearly, we have a V \(W {}) = c V \(W {}) + γ. Since g = (v p+, v p+2,, v k ) i a firing equence of a, g i alo a firing equence of c V \(W {}) + γ in G/W {}. It follow from Lemma 5 that c V \(W {}) i a recurrent configuration of G/W {} with repect to ink t. For the convere tatement let h = (u, u 2,, u p ) be a firing equence of d uch that d + γ h d in G/W {}, then u i W {} for any i, and each vertex of G not in W {} occur exactly once in h. Let b be uch that c + β w,w 2,,w q b in G. Clearly, b V \(W {}) = d + γ, therefore (w, w 2,, w q, u, u 2,, u p ) i a firing equence of c + β in G. It follow that c C. Lemma 3. Let e and e be two revere arc of G uch that they are not loop and e =. Let H denote G\{e, e } and w denote e +. If H i connected then {c C : c(w) < deg + (w) } i the et of all recurrent configuration of H with repect to ink. the electronic journal of combinatoric 23() (206), #P.57 22

Proof. We prove a double incluion, uing again both direction of Lemma 5 (Burning algorithm). Let β : V \{} N be given by β(v) = deg G (, v) for any v V \{}, and γ : V \{} N be given by γ(v) = deg H (, v) for any v V \{}. We have β(v) = γ(v) for any v w, and β(w) γ(w) =. Let c C uch that c(w) < deg + (w). We have to prove that c i alo a recurrent configuration of H with repect to ink. Let f = (v, v 2,, v k ) be a firing equence of c in G uch that v i for any i, c + β f c, and each vertex of G ditinct from occur exactly once in f. We will how that f i a firing equence of c + γ in H. Let j be uch that v j = w. Clearly, (v, v 2,, v j ) i a firing equence of c + β and c + γ in G and H, repectively. Let a be uch that c + β v,v 2,,v j a in G and b be uch that c + γ v,v 2,,v j b in H. It follow from β(w) γ(w) = that a(w) b(w) =. To prove that f i a firing equence of c + γ in H it uffice to how that v j i firable in b with repect to H. Since v j i firable in a with repect to G, we have a(w) deg + (w), therefore b(w) deg + (w). It follow that w i firable in b with repect to H. Thi implie that f i alo a firing equence of c + γ with repect to H. By Lemma 5, c i a recurrent configuration of H with repect to ink. For the convere, let d be a recurrent configuration of H with repect to ink. Let g = (u, u 2,, u p ) be a firing equence of d + γ in H uch that u i for any i, d + γ g d in H, and each vertex of H ditinct from occur exactly once in g. We have d(w) deg + (w) < deg + (w). By imilar argument a above, g i alo a firing equence of d + β in G, therefore d i a recurrent configuration of G with repect to ink. Firt, the Tutte polynomial on undirected graph ha the recurive formula T G (, y) = y T G\e (, y) if e i a loop. We have the following generalization. Propoition 4. If e i a loop then T G (y) = y T G\e (y). Proof. Let denote e. Let C be the et of all recurrent configuration of G with ink. Clearly, C i alo the et of all recurrent configuration of G\e with ink. Since deg + () deg + () =, for any c C we have level G (c) level H (c) =. Thi implie that T G (y) = yt G\e (y). Second, in order to generalize the recurive formula T G (, y) = T G/e (, y) if e i a bridge, we recall the definition of trong bridge of a directed graph. Definition. Let H be a directed graph. An arc a of H i called trong bridge if H\a ha more trongly connected component than H. The next lemma i ued in the proof of ubequent propoition. It aim at howing that there i a imilarity between a trong bridge of an Eulerian graph and a bridge of an undirected graph. Lemma 4. Let a be a trong bridge of G. Then there i a ubet X of V uch that {a} = {e A : e X, e + X}. Moreover, there i an arc b in G uch that {b} = {e A : e X, e + X}. the electronic journal of combinatoric 23() (206), #P.57 23