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1 Les Cahers du GERAD ISSN: Countng the Number of Non-Equvalent Vertex Colorngs of a Graph A. Hertz H. Mélot G November 2013 Les textes publés dans la sére des rapports de recherche HEC n engagent que la responsablté de leurs auteurs. La publcaton de ces rapports de recherche bénéfce d une subventon du Fonds de recherche du Québec Nature et technologes.

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3 Countng the Number of Non-Equvalent Vertex Colorngs of a Graph Alan Hertz GERAD & Polytechnque Montréal Montréal (Québec) Canada, H3A 3A7 alan.hertz@gerad.ca Hadren Mélot Algorthms Lab Unversté de Mons B-7000 Mons, Belgum hadren.melot@umons.ac.be November 2013 Les Cahers du GERAD G Copyrght c 2013 GERAD

4 G Les Cahers du GERAD Abstract: We study the number P(G) of non-equvalent ways of colorng a gven graph G. We show some smlartes and dfferences between ths graph nvarant and the well known chromatc polynomal. Relatons wth Strlng numbers of the second knd and wth Bell numbers are also gven. We then determne the value of ths nvarant for some classes of graphs. We fnally study upper and lower bounds on P(G) for graphs wth fxed maxmum degree. Key Words: Non-equvalent colorngs, number of colorngs, chromatc polynomal. Résumé : Nous étudons le nombre P(G) de coloratons non-équvalentes des sommets d un graphe G. Nous montrons quelques smlartés et dfférences entre ce nouvel nvarant et le très célèbre polynôme chromatque. Nous ndquons également quelques relatons ntéressantes entre cet nvarant et les nombres de Strlng du deuxème type et les nombres de Bell. Nous détermnons ensute la valeur de cet nvarant pour quelques classes de graphes. Fnalement, nous présentons des bornes nféreures et supéreures sur P(G) pour des graphes ayant un degré maxmum fxé. Acknowledgments: Ths work was ntated durng a vst partly funded by the F.R.S.-FNRS of Alan Hertz at the Algorthms Lab (Unversté de Mons).

5 Les Cahers du GERAD G Introducton A queston whch probably sounds famlar for many researchers n graph theory s: what s the number of ways of colorng a gven graph G? For the path P 3 on three vertces, an answer that makes sense s two as depcted n Fgure 1. Indeed, at least two colors are needed, and there s only one colorng wth two colors (the two extremtes share the same color whle the central vertex has ts own color), and only one colorng wth three colors (each vertex has ts own color). a b a a b c Fgure 1: The 2 non-equvalent colorngs of P 3 (usng any number of colors). However, snce more than 100 years, the common answer to the above queston for P 3 s not two but twelve. To understand why, we recall the noton of chromatc polynomal whch was ntroduced by Brkhoff n an attempt to prove the four-color theorem. In a paper publshed n 1912 [1] Brkhoff 1 proves that The number of ways of colorng a gven map M n k colors (k = 1, 2,...) s gven by a polynomal P (k) of degree n, where n s the number of regons n the map M. Brkhoff started the study of ths polynomal by defnng a quantty m as the number of ways of colorng the map by usng exactly colors when mere permutatons of the colors are dsregarded. Then, he used ths quantty to defne m k! (k )! as the number of ways of colorng the gven map n exactly of the k colors, countng two colorngs as dstnct when they are obtaned by a permutaton from the other. Denotng G the planar graph correspondng to the map M, we can therefore defne Π(G, k) = k =1 m k! (k )! as the number of ways of colorng G wth at most k colors, countng two colorngs as dstnct when they are obtaned by a permutaton from the other. The same defnton also apples for non-planar graphs G. The chromatc polynomal s the polynomal of degree n passng by ponts (k, Π(G, k)) for k = 0, 1,..., n. For example, for the path P 3 we have Π(P 3, k) = k(k 1) 2. Indeed, Π(P 3, 0) = Π(P 3, 1) = 0; Π(P 3, 2) = 2 (take for nstance the frst two colorngs n the left column of Fgure 2) and Π(P 3, 3) = 12 as shown n Fgure 2. The number of vertex colorngs of a graph G s nowadays commonly nterpreted as Π(G, n), where n s the number of vertces n G. However, we argue that the quantty m defned by Brkhoff,.e., the number of non-equvalent colorngs wth an exact number of used colors s also of nterest. Ths s especally the case when a set of elements has to be parttoned nto a gven number of non-empty subsets, subject to some constrants. In the next secton we fx some notatons and gve a formal defnton of the number P(G) of non-equvalent vertex colorngs of a graph G. Smlartes and dfferences between ths nvarant and the chromatc polynomal are studed n Secton 3. In Secton 4, we address the problem of computng P(G), and we gve exact values for some partcular graphs n Secton 5. Then, n Secton 6, we prove some bounds on P(G) for graphs of bounded maxmum degree and let other bounds as open problems. 1 We replaced λ used n [1] by k to unfy notatons, even n the quotes from ths paper.

6 2 G Les Cahers du GERAD a b a a b c b a b a c a c a c b c b c b c a c b b a c b c a c a b c b a Fgure 2: The 12 colorngs of P 3 (usng 3 colors) as defned by the chromatc polynomal. 2 Notatons For basc notons of graph theory that are not defned here, we refer to Destel [3]. Let G = (V, E) be a smple undrected graph. We denote by n = V the order of G and by m = E ts sze. We wrte G H f G and H are two somorphc graphs. Let K n (resp. C n and P n ) be the complete graph (resp. the cycle and the path) of order n. The wheel W n s the graph of order n obtaned by connectng a vertex to all vertces of C n 1. Also, we wrte K a,b for the complete bpartte graph where a and b are the cardnaltes of the two sets of vertces of the bpartton. Fnally, let S n be the star on n vertces, that s K 1,n 1. Let N(v) denote the neghbors of a vertex v n G. Vertex v s sad to be smplcal f N(v) nduces a clque n G. A graph s chordal (or trangulated) f every cycle of length larger than 3 has a chord. The degree of a vertex v s denoted d(v) (.e., d(v) = N(v) ). A vertex v s solated f d(v) = 0 and s domnatng f d(v) = n 1. The maxmum degree of G s denoted (G). Let u and v be two vertces n a graph G of order n, we denote G \ uv the graph (of order n 1) obtaned by dentfyng (mergng) the vertces u and v and, f uv E(G), by removng edge uv. Also, f uv E(G), we note G uv the graph obtaned from G by removng edge uv, whle f uv / E(G), the graph G + uv s the graph obtaned by addng uv n G. For a vertex v of G, we denote G v the graph obtaned from G by removng v and all ts ncdent edges. A vertex colorng (or smply a colorng n the sequel) s an assgnment of colors to the vertces of G. A proper colorng s a colorng such that adjacent vertces have dfferent colors. The chromatc number χ(g) of a graph G s the mnmum numbers of colors n a proper colorng of G. Two colorngs are equvalent f they nduce the same partton of the vertex set. We defne P (G, k) as the number of proper non-equvalent colorngs of a graph G that use exactly k colors. The total number P(G) of non-equvalent colorngs of a graph G s then defned as: n P(G) = P (G, k). (1) k=χ(g) As mentoned n the prevous secton, Π(G, k) s the number of proper colorngs of a graph G that use at most k colors, countng two non-dentcal colorngs as dstnct when they are obtaned by a permutaton from the other.

7 Les Cahers du GERAD G Smlartes and Dfferences Accordng to ther defntons, Π(G, k) and P (G, k) are lnked wth the followng relatons: Π(G, k) = k j=χ(g) k! P (G, j), (2) (k j)! and Π(G, k) kπ(g, k 1) P (G, k) =. (3) k! Observe that unlke P (G, k), Π(G, k) also counts colorngs wth strctly less than k colors. Moreover, whle P(G) and Π(G, n) mght appear as smlar concepts (snce they both count colorngs wth at most n colors), they dffer n varous ways. We have already mentoned that only non-equvalent colorngs are counted n P(G), whch means that P(G) corresponds to the number of parttons of the vertex set of G, takng nto account constrants that prevent some pars of vertces of belongng to the same subset of the partton. To accentuate these dfferences, observe that f Π(G, n) < Π(H, n) for two graphs G and H of order n, ths does not necessarly mply that P(G) < P(H) (and conversely) as shown n Fgure 3. P(G) = 18 Π(G, 6) = 8520 P(H) = 17 Π(H, 6) = 9000 Fgure 3: Two graphs G and H wth 6 vertces such that Π(G, 6) < Π(H, 6) and P(G) > P(H). Also, there exst pars of graphs (G, H) such that P(G) = P(H) but Π(G, n) Π(H, n), and conversely (see examples n Fgure 4). P(G) = 4 Π(G, 5) = 420 P(G ) = 6 Π(G, 5) = 600 P(H) = 4 Π(H, 5) = 480 P(H ) = 5 Π(H, 5) = 600 Fgure 4: Two pars of graphs wth 5 vertces showng that equalty for one way to counts the colorngs does not mply equalty for the other. There are also dfferences at a computatonal level. Gvng a graph G wth a domnatng vertex v, the followng property states that the computaton of P(G) can be reduced to that of P(G v). A smlar trval reducton does not hold for Π(G). Property 1 If a graph G has a domnatng vertex v, then P(G) = P(G v).

8 4 G Les Cahers du GERAD Proof. Snce v s a domnatng vertex, t must have ts own color n all colorngs of G, whch means that the number of proper non-equvalent colorngs of G remans the same when v s removed. Conversely, f G s the dsjont unon of two graphs G 1 and G 2, t s easy to compute Π(G 1 G 2 ) by takng product of Π(G 1 ) and Π(G 2 ). However, the followng property shows that the computaton of P(G 1 G 2 ) s more ntrcate. Property 2 Let G = G 1 G 2 be a graph that s the dsjont unon of two graphs G 1 and G 2. Then, P(G) = n k k=1 =1 j=0 ( P (G 1, )P (G 2, k j) j )( ) k j ( j)! j Proof. The frst sum on k comes smply from the defnton (1) of P(G). The two nner sums compute P (G, k) as follows. Let k be the number of colors used for G 1. Let j bet an nteger such that j represents the number of colors that are used both n G 1 and n G 2. The value of j can vary from 0 (that s colors are shared) to (that s no color are shared). Observe that n order to use exactly k colors for G, G 2 must be colored wth exactly k j colors. Fnally, the term ( j) counts the numbers of ways to choose the j shared colors nto G 1, the term ( k j j) does the same for G2 and ( j)! counts all the possble permutatons for ths shared colors. As a corollary, we get the followng result whch wll be useful n later sectons. Corollary 3 Let G = K p K q be the dsjont unon of two clques of szes p and q such that p q. Then, p+q P(G) = k=q ( p )( k q q p + q k ) (p + q k)! Proof. We apply Property 2 knowng that P(K p, ) = 1 f and only f = p and P(K q, k j) = 1 f and only f j = k q. For all other values of and j, the nner products beng equal to zero. Also, observe that f k < q, there are not enough colors for a proper colorng of G. 4 Countng the colorngs recursvely As for several other algorthms n graph colorng, the deleton-contracton rule s a well known method to compute the chromatc polynomal. More precsely, we have: where uv s any edge of G, and Π(G, k) = Π(G uv, k) Π(G \ uv, k), Π(G, k) = Π(G + uv, k) + Π(G \ uv, k), for any par of dstnct vertces u and v such that uv / E(G). These recurrences, whch are often called the Fundamental Reducton Theorem [4], are also vald to compute P (G, k) and P(G). More precsely, let u and v be any par of dstnct vertces of G, we have, P (G, k) = P (G uv, k) P (G \ uv, k), (4) f uv E(G), and P (G, k) = P (G + uv, k) + P (G \ uv, k), (5) f uv / E(G). Smlarly, f uv E(G), we have,

9 Les Cahers du GERAD G P(G) = P (G uv) P (G \ uv), (6) and f uv / E(G). P(G) = P (G + uv) + P (G \ uv), (7) Snce there s only one possble colorng for K n (usng exactly n colors), we have { 1 f k = n, P (K n, k) = 0 otherwse, and P(K n ) = 1. Ths consttutes a base case for a straghtforward recursve algorthm to compute P(G) for any graph G usng relaton (7). Another recursve procedure can be obtaned from (6) usng the empty graph K n to defne the base case. Indeed, we have { { n } k k n, P (K n, k) = 0 k > n, where { } n = 1 k k! k ( ) k ( 1) k j j n j j=1 s a Strlng number of the second knd, that s the number of ways to partton a set of n elements nto k non-empty subsets. It follows that n { } n P(K n ) = = B n, k k=1 where B n s the n th Bell number (sequence A n OEIS [8]). Ths s not surprsng snce B n represents the number of parttons of a set of n elements whch s obvously the same as the number of non-equvalent colorngs n a graph wthout any edge. Of course, the complextes of the two above recursve algorthms are exponental n general. However, we wll see n the next secton that t can be refned to gve a polynomal algorthm for some partcular classes of graphs. Generalzed Strlng and Bell numbers have been defned and studed n [2] and are also lnked to the new proposed nvarant. More precsely, let S r (n, k) = 1 k! k ( ) ( ) n k j! ( 1) k j. j (j r)! j=r Consder n sets E 1, E 2,..., E n of r elements. The generalzed Strlng number S r (n, k) s the number of dfferent parttons of these nr elements nto k non-empty subsets such that each subset contans at most one element of each E. In other words, S r (n, k) = P (nk r, k). The Generalzed Bell numbers B r,n are then defned as follows: B r,n = rn k=r S r (n, k). They represent the number of parttons of the nr elements so that each subset contans at most one element of each E. Hence, B r,n = P(nK r ).

10 6 G Les Cahers du GERAD 5 Determnng some numbers of colorngs In ths secton, we determne the value of P(G) for several classes of graphs. We start wth k-trees whch are chordal graph wth all maxmal clques of sze k + 1 and all mnmal clque separators of sze k. Thus, a 1-tree s a tree. Note that every k-tree can be constructed from a complete graph on k + 1 vertces by addng vertces teratvely such that each new vertex has exactly k neghbors formng a clque [7]. To avod confuson wth the number of colors k we use the notaton r-tree n the sequel. Theorem 4 Let Tn r be a r-tree of order n r + 1. Then, { } n r P (Tn, r k) =, k r for all k = r + 1,..., n. Proof. If n = r + 1, then T r n s a clque of sze r + 1 and P (T r n, n) = 1 = { n r}. Otherwse, let v be a smplcal vertex of T r n. We consder two cases when countng the colorngs of T r n: ether v has ts own color, or v has one color already used by other vertces of T r n. The frst case gves P (T r n 1, k 1) colorngs and the latter one (k r)p (T r n 1, k) colorngs snce the color of v cannot be the same than the r colors already used by ts neghbors. Altogether we have P (T r n, k) = P (T r n 1, k 1) + (k r)p (T r n 1, k). Usng nducton, we get P (Tn, r k) = { } { n 1 r k 1 r + (k r) n 1 r } { k r = n r k r}. The last equalty comes from the known recurrence relatons obeyed by Strlng numbers. Theorem 5 Let T r n be a r-tree of order n r + 1. Then, P(T r n) = B n r. Proof. By Theorem 4, and snce any colorng of T r n has at least r + 1 colors, P(Tn) r = n k=r+1 P (T n, r k), = n = n r k=1 { n r k=r+1 k r}, { n r k } = Bn r. Note that f r = 0, then r-trees are empty graphs and Theorem 5 s another way to show that P(K n ) = B n. Another nterestng partcular case of r-trees are trees. Corollary 6 Let T be a tree of order n 1. Then, P(T ) = B n 1. The decomposton used n the proof of Theorem 4 allows to compute the number of non-equvalent colorngs of a chordal graph n polynomal tme (usng dynamc programmng). Indeed, f v s a smplcal vertex of G wth r neghbors, then P (G, k) = P (G v, k)(k r) + P (G v, k 1). Corollary 7 If G s a chordal graph, then there exsts a polynomal algorthm to compute P(G).

11 Les Cahers du GERAD G Notce the same results also holds for the computaton of the chromatc polynomal [6]. Theorem 8 Let K 2,n be a complete bpartte graph on n + 2 vertces such that V = V 1 V 2 and V 1 = 2. Then, P(K 2,n ) = 2B n. Proof. Let H be the graph obtaned from K 2,n wth an addtonal edge between the two vertces of V 1. Observe that H has a domnatng vertex. Thus, applyng (5), and then Property 1 and Corollary 6 gves P(K 2,n ) = P(K 1,n ) + P(H) = 2P(K 1,n ) = 2B n. Theorem 9 Let C n be a cycle of order n 3. Then, n 1 P(C n ) = ( 1) j+1 B n j. Proof. Note that the result holds for C 3 = K 3 snce B 2 B 1 = 1. Applyng (6) gves Then, by Corrolary 6 and the nducton on n, j=1 P(C n ) = P(P n ) P(C n 1 ). P(C n ) = B n 1 n 2 j=1 ( 1)j+1 B n 1 j, = B n 1 n 1 j=2 ( 1)j B n j, = B n 1 + n 1 j=2 ( 1)j+1 B n j, = n 1 j=1 ( 1)j+1 B n j. Corollary 10 Let W n be a wheel of order n 4. Then, n 2 P(W n ) = ( 1) j+1 B n j 1. Proof. By Property 1, we have P(W n ) = P(C n 1 ), and the result follows from Theorem 9. j=1 Observe that Bell numbers appear repeatedly n the above results. Recall that B n s the number of parttons of a set of n labeled elements wthout any constrant on the fact that two elements can be n the same partton or not. From a graph theoretcal pont of vew, the parttons are the colors of the vertces and addng an edge represents such a constrant. In partcular, t s of nterest to note that the sequence P(C n ) for n = 2, 3,... determned by Theorem 9 corresponds to sequence A n OEIS [8]. Ths sequence s known to be the number of cyclcally spaced parttons. Gven two graphs G and H of order n, we note G > P H and say that G strctly domnates H for the number of non-equvalent colorngs f P (G, k) P (H, k) for all k = 1, 2,..., n, and there exsts some nteger k such that P (G, k) > P (H, k). By Property 2, the followng corollary s straghtforward. Corollary 11 Let G, G and H be three graphs such that G and G have the same order. If G > P G, then, P(G H) > P(G H).

12 8 G Les Cahers du GERAD 6 Boundng the number of colorngs of graphs wth fxed maxmum degree In ths secton, we study upper and lower bounds on P(G) for graphs G wth bounded maxmum degree. We note that the followng results were frst conjectured wth the help of the conjecture-makng system GraPHedron [5]. The upper bound s straghtforward. We defne G > n, to be the graph of order n and wth a maxmum degree that s composed of a star S +1 and n 1 solated vertces (see Fgure 5 for an example). Fgure 5: The graph G > 8,4. Theorem 12 Let G be a graph of order n and maxmum degree. Then, ( ) P(G) ( 1) B n, =0 wth equalty f and only f G s somorphc to G > n,. Proof. The graph G > n, s clearly the graph mnmzng the number of edges among all graphs of order n wth maxmum degree. Addng edges to G > n, (n such a way that the maxmum degree s not ncreased) wll add new constrants between pars of vertces, and ths wll therefore strctly decrease the number of colorngs. Hence P(G) P(G > n, ), wth equalty f and only G s somorphc to G> n,. It remans to prove that P(G > n, ) = ( ) ( 1) B n for all n and. =0 The equalty holds for = 0 snce P(G > n, ) s then somorphc to K n and we have already observed that P(K n ) = B n. For larger values of, we proceed by nducton usng the followng equalty obtaned from (6): P(G > n, ) = P(G> n, 1 ) P(G> n 1, 1 ). We then have P(G > n, ) = 1 =0 ( 1)( 1 = 1 =0 ( 1)( 1 ) Bn 1 ) Bn + = B n + 1 =1 ( 1) ( ( 1 = =0 ( 1)( ) Bn. =0 ( 1)( ) 1 Bn 1 =1 ( 1)( ) 1 1 Bn ) ( + 1 ) 1 ) + ( 1) B n A lower bound on P(G) for graphs of order n and bounded maxmum degree s easy to obtan for some values of, but more ntrcate or stll open for the other ones. In the rest of ths secton, we say that a graph G s extremal f P(G ) P(G) for all graphs G of order n such that (G) = (G ). The followng property wll be used ntensvely n the ongong proofs. Property 13 Let G be a graph wth two vertces v and w such that vw / E and Then, G s not extremal. max(d(v), d(w)) < (G).

13 Les Cahers du GERAD G Proof. Addng the edge vw wll not change the value of (G) but wll strctly decrease the number of colorngs of G. We start by defnng a graph of order n and wth maxmum degree equals to 1. If n s even, then G < n, =1 s the dsjont unon of n 2 copes of K 2; f n s odd, t s the dsjont unon of G < n 1, =1 and an solated vertex. The graph G < 7, =1 s drawn on the left-hand sde of Fgure 6. Fgure 6: The graphs G < 7, =1, G< 7, =2 and K 6 K 1 (from left to rght). Theorem 14 Let G be a graph of order n such that (G) = 1. Then, P(G) n/2 =0 wth equalty f and only f G s somorphc to G < n, =1. ( 1) ( n/2 ) B n, Proof. Snce (G) = 1, G s a dsjont unon of several copes of K 2 and solated vertces. If G has at least two solated vertces v and w, we know from Property 13 that t cannot be extremal. Thus, f G s extremal t must be somorphc to G < n, =1. Consder now the dsjont unon of p K 2 and q K 1. We prove that p ( ) p P(pK 2 qk 1 ) = ( 1) B 2p+q. =0 The equalty holds for p = 0 snce the graph s then somorphc to K q and we have P(K q ) = B q. For larger values of p, we proceed by nducton usng the followng equalty obtaned from (6): We then have P(pK 2 qk 1 ) = P((p 1)K 2 (q + 2)K 1 ) P((p 1)K 2 (q + 1)K 1 ). P(pK 2 qk 1 ) = p 1 =0 ( 1)( p 1 = p 1 =0 ( 1)( p 1 ) B2p+q p 1 ) B2p+q + p = B 2p+q + p 1 =1 ( 1) ( ( p 1 = p =0 ( 1)( p ) B2p+q. =0 ( 1)( ) p 1 B2p+q 1 =1 ( 1)( p 1 1) B2p+q ) ( + p 1 1) ) + ( 1) p B p+q To conclude, t s suffcent to observe that G < n, =1 s somorphc to pk 2 qk 1 wth p = n/2 and q = n 2p. We now consder graphs G wth maxmum degree (G) = 2. Before gvng a lower bound on P(G) for such graphs, we prove some useful lemmas. Lemma 15 Consder a cycle C n of order n 6. Then, P (C n, k) > P (C n 3 C 3, k) for k = 3, 4,..., n 2; P (C n, k) = P (C n 3 C 3, k) for k = n 1, n.

14 10 G Les Cahers du GERAD Proof. The values n the followng table show that the result holds for n = 6. k P (C 6, k) P (2C 3, k) For larger values or n, the followng equaltes are obtaned from (4)and (5): P (C n 3 C 3, k) = P (P n 3 C 3, k) P (C n 4 C 3, k) = P (P n 3 P 3, k) P (P n 3 P 2, k) P (C n 4 C 3, k) = (P (P n, k) + P (P n 1, k)) (P (P n 1, k) + P (P n 2, k)) P (C n 4 C 3, k) = P (P n, k) P (P n 2, k) P (C n 4 C 3, k) Clearly, P (P n 2, k) > 0 for k = 3, 4,..., n 2 and P (P n 2, k) = 0 for k = n 1, n. Also, by nducton, we have P (C n 4 C 3, k) < P (C n 1, k) for k = 3, 4,..., n 2, and P (C n 4 C 3, k) = P (C n 1, k) for k = n 1, n. Hence, P (C n 3 C 3, k) P (P n, k) P (C n 1, k), wth equalty only f k = n 1, n. To conclude, we observe from (4) that P (P n, k) P (C n 1, k) = P (C n, k). Snce P (C n, 2) 0 whle P (C n 3 C 3, 2) = 0 for n 6, the followng corollary s straghtforward. Corollary 16 Consder a cycle C n of order n 6. Then C n > P C n 3 C 3. Lemma 17 Consder a cycle C n of order n 3. Then, P (C n K 1, k) = P (P n+1, k) for k = 3, 4,..., n + 1. Proof. The result s vald for n = 3 snce P (C 3 K 1, 3) = P (P 4, 3) = 3 and P (C 3 K 1, 4) = P (P 4, 4) = 1. For larger values or n and k 3, we proceed by nducton and apply (4) and (5) to obtan: P (C n K 1, k) = P (P n K 1, k) P (C n 1 K 1, k) = P (P n+1, k) + P (P n, k) P (C n 1 K 1, k) = P (P n+1, k) Corollary 18 Consder a cycle C n of order n 4. Then C n K 1 > P C n+1 C n K 1 > P C n 2 C 3 f n s even; f n s odd. Proof. Snce P (P n+1, k) > P (C n+1, k) for k = 3, 4,..., n, t follows from Lemma 17 that P (C n K 1, k) > P (C n+1, k) for k = 3, 4,..., n. If n s even, then P (C n K 1, 2) = 2 > 0 = P (C n+1, 2) and P (C n K 1, n + 1) = P (C n+1, n + 1) = 1, whch mples C n K 1 > P C n+1. If n s odd, then we know from Lemma 15 that P (C n+1, k) P (C n 2 C 3, k) for k = 3, 4,..., n. Snce P (C n K 1, 2) = P (C n 2 C 3, 2) = 0 and P (C n K 1, n + 1) = P (C n 2 C 3, n + 1) = 1, we have C n K 1 > P C n 2 C 3. Lemma 19 Consder a cycle C n of order n 5. Then, C n 2 K 2 > P C n.

15 Les Cahers du GERAD G Proof. By applyng (4) and (5), we obtan the followng equaltes whch are vald for all k 2: P (C n 2 K 2, k) = P (P n 2 K 2, k) P (C n 3 K 2, k) = P (P n, k) + P (P n 1, k) P (P n 3 K 2, k) +P (C n 4 K 2, k) = P (P n, k) + P (P n 1, k) P (P n 1, k) P (P n 2, k) +P (C n 4 K 2, k) = P (P n, k) P (P n 2, k) + P (C n 4 K 2, k). (8) We now analyse three dfferent cases. If k 4, we frst show that P (C n 2 K 2, k) = P (P r, k). Ths s true for n = 5, 6 snce P (C 3 K 2, 4) = P (P 5, 4) = 6, P (C 3 K 2, 5) = P (P 5, 5) = 1, P (C 4 K 2, 4) = P (P 6, 4) = 25, P (C 4 K 2, 5) = P (P 6, 5) = 10, and P (C 4 K 2, 6) = P (P 6, 6) = 1. For larger values or n, the equalty s obtaned by nducton, usng equaton (8), snce P (C n 4 K 2, k) s then equal to P (P n 2, k). Snce P (C n, k) P (P n, k) for all k, we have P (C n, k) P (C n 2 K 2, k) for k = 4, 5,..., n. If k = 3, we frst show that P (C n 2 K 2, 3) = P (P n, 3) + ( 1) n. Ths s true for n = 5, 6 snce P (C 3 K 2, 3) = 6, P (P 5, 3) = 7, P (C 4 K 2, 3) = 16, and P (P 6, 3) = 15. For larger values or n, the equalty s obtaned by nducton, usng equaton (8), snce P (C n 4 K 2, k) s then equal to P (P n 2, 3)+ ( 1) n 2 = P (P n 2, 3) + ( 1) n. Snce P (C n 1, 3) > 1 for all n 5, we conclude that P (C n, 3) = P (P n, 3) P (C n 1, 3) P (P n, 3) 2 < P (P n, 3) + ( 1) n = P (C n 2 K 2, 3). If k = 2 then P (C n, 2) P (C n 2 K 2, 2) snce both P (C n, 2) and P (C n 2 K 2, 2) equal 0 f n s odd, whle P (C n, 2) = 1 < 2 = P (C n 2 K 2, 2) f n s even. The graph G < n, =2 s defned as follows: t s the dsjont unon of n 3 copes of K 3 f n 0 (mod 3); t s the dsjont unon of G < n 4, =2 and C 4 f n 1 (mod 3); t s the dsjont unon of G < n 5, =2 and C 5 f n 2 (mod 3). The graph G < 7, =2 s llustrated n the mddle of Fgure 6. We now gve a lower bound on P(G) for graphs G wth maxmum degree (G) = 2 and order n 5. Ths s not restrctve because f n 4 and = 2, then = n 2 or = n 1 and these cases are treated later. Theorem 20 Let G be a graph of order n 5 such that (G) = 2. Then, P(G < n, =2 ) P(G), wth equalty f and only f G s somorphc to G < n, =2. Proof. Suppose G s extremal. Snce (G) = 2, G s the dsjont unon of cycles and paths. It follows from Property 13 that at most one connected component of G s a path, and such a path can only be K 1 or K 2. Case 1: K 1 s a connected component of G. Let C r (r 3) be a longest cycle of G. If r = 3, then G s the dsjont unon of K 1 and at least two copes of C 3 (because n 5). Thus, G = 2C 3 K 1 H where H s a (possbly empty) dsjont unon of C 3. The followng table shows that G s not extreme snce 2C 3 K 1 > P C 3 C 4, a contradcton.

16 12 G Les Cahers du GERAD k P (K 1 2C 3, k) P (C 3 C 4, k) If r 4, then we know from Corollary 18 that ether C r+1 (f r s even) or C r 2 C 3 (f r s odd) s strctly domnated by C r K 1. Hence, G s not extremal, a contradcton. Case 2: K 2 s a connected component of G. Let C r be any cycle n G. We know from Lemma 19 that C r K 2 > P extremal, a contradcton. C r+2, whch means that G s not Case 3: G s the dsjont unon of cycles. Snce G s extremal, we know from Corollary 16 that these cycles are copes of C 3, C 4 or C 5. The followng tables show that 2C 5 > P 2C 3 C 4, C 5 C 4 > P 3C 3, and 2C 4 > P 2C 5 C 3. Hence, snce G s extremal, t contans no more than one C 4 or one C 5, whch means that G s somorphc to G < n, =2. k P (2C 5, k) P (2C 3 C 4, k) k P (C 5 C 4, k) P (3C 3, k) k P (2C 4, k) P (C 5 C 3, k) Snce C 3 = K 3, we can lnk the above result wth the generalzed Bell numbers mentoned n Secton 4. Corollary 21 Let G be a graph of order n such that n 0 (mod 3) and (G) = 2. Then P(G) B 3, n 3 We now gve a lower bound on P(G) for graphs G of order n and maxmum degree n 2. Theorem 22 Let G be a graph of order n 2 such that (G) = n 2. Then, P(G) n wth equalty f and only f G s somorphc to K n 1 K 1 when n 4, and G s somorphc to K 3 K 1 or C 4 otherwse. Proof. The proof s by nducton on n and the result s clearly vald for n = 2. Notce frst that P(K n 1 K 1 ) = n because ether the solated vertex of K 1 has ts own color, or t uses one of the n 1 colors n K n 1. So let G be an extremal graph of order n > 2 wth (G) = n 2. We then have P(G) P(K n 1 K 1 ) = n. Let x be any vertex of degree n 2, and let y be the unque vertex that s not adjacent to x. It follows from Property 13 that f two vertces v and w dstnct from x and y are non-adjacent, then they are both adjacent to y. Hence, f y s an solated vertex n G, then G s somorphc to K n 1 K 1.

17 Les Cahers du GERAD G So suppose d(y) 1 and let v be one of ts neghbors. Snce v s not domnatng, there exsts at least one vertex w not adjacent to v. As observed above, w s necessarly adjacent to y. Let W be the set of vertces adjacent to y. We therefore have W 2 and, by Property 13, every vertex non-adjacent to y has degree n 2. Let G be the graph nduced by W. No vertex of G s domnatng (else t would also be domnatng n G), and snce at least one of v and w has degree n 2 n G (and thus has degree W 2 n G ), we have (G ) = W 2. By nducton, P(G ) W. Gven any colorng of G, we can construct n W non-equvalent colorngs of G by copyng the colors on the vertces of W, assgnng new colors to all vertces non-adjacent to y, and ether assgnng one of these n W 1 new colors to y, or a new one not shared by any other vertex. Hence, n P(G) P(G )(n W ) W (n W ). (9) Then, n W W (n W 1) 2(n W 1), whch mples n W 2. Snce x and y do not belong to W, we have n W = 2. Hence, equaton (9) becomes W W, whch s equvalent to W 2. Snce v and w belong to W, we have W = 2. In summary, P(G) = n = 4 and G s somorphc to C 4. Fnally, notce that the lower bound on P(G) for graphs G wth (G) = n 1 s trval snce K n has clearly the mnmum number of colorngs among all graph of order n. 7 Concludng remarks and open problems We have defned a graph nvarant that corresponds to the number of non-equvalent proper vertex colorngs of a graph. We have shown smlartes and dfferences between ths nvarant and the famous chromatc polynomal. We have also determned the value of ths nvarant for several classes of graphs and have gven lower and upper bounds on ts value for graphs wth bounded maxmum degree. It would be nterestng to determne a lower bound on P(G) for graphs G of order n and wth maxmum degree n {3, 4,..., n 3}. The extremal graphs n ths case do not seem to have a smple structure, as was the case for (G) = 1, 2, n 2, n. We have determned some of them by exhaustve enumeraton. For example, we have drawn n Fgure 7 the only graphs G of order n = 6, 7, 8, 9 wth mnmum value P(G) when (G) = 3, 4, 5. Notce also that several graphs wth mnmum value P(G) are non-connected. It would be nterestng to determne these extremal graphs wth the addtonal constrant that G must be connected. Also, t could be nterestng to characterze the graphs G that mnmze or maxmze P(G) when the order and the sze of G are fxed.

18 14 G Les Cahers du GERAD (G) = 3 (G) = 4 (G) = 5 n = 6 P(G) = 18 P(G) = 6 P(G) = 1 n = 7 P(G) = 70 P(G) = 29 P(G) = 7 n = 8 P(G) = 209 P(G) = 106 P(G) = 43 n = 9 P(G) = 1274 P(G) = 456 P(G) = 202 Fgure 7: Unque graphs G of order n = 6, 7, 8, 9 and maxmum degree (G) = 3, 4, 5 wth mnmum value for P(G). References [1] Brkhoff, G.D. A Determnant Formula for the Number of Ways of Colorng a Map. Ann. Math. Harvard Coll. 14 (1912), [2] Blasak, P., Penson, K.A., Solomon, A.I. The Boson Normal Orderng Problem and Generalzed Bell Numbers Annals of Combnatorcs 7 (2003), [3] Destel, R. Graph Theory, second edton, Sprnger-Verlag, [4] Dong, F.M., Koh, K.M., Teo, K.L. Chromatc Polynomals and Chromatcty of Graphs, World Scentfc Publshng Company, 2005, ISBN [5] Mélot, H. Facet defnng nequaltes among graph nvarants: the system GraPHedron. Dscrete Appled Mathematcs 156 (2008), [6] Naor, J., Naor, M., Schaffer, A. Fast parallel algorthms for chordal graphs. Proc. 19th ACM Symp. Theory of Computng (1987), [7] Patl, H.P. On the structure of k-trees. Journal of Combnatorcs, Informaton and System Scences 11 (1986), [8] Sloane, N. The on-lne encyclopeda of nteger sequences. Avalable at sequences/.