Acyclic Colorings of Directed Graphs

Transcription

1 Acyclic Colorings of Directed Graphs Noah Golowich Septeber 9, 014 arxiv: v1 [ath.co] 6 Sep 014 Abstract The acyclic chroatic nuber of a directed graph D, denoted χ A (D), is the iniu positive integer k such that there exists a decoposition of the vertices of D into k disjoint sets, each of which induces an acyclic subgraph. For any 1, we introduce a generalization of the acyclic chroatic nuber, naely χ (D), which is the iniu nuber of sets into which the vertices of a digraph can be partitioned so that each set is weakly -degenerate. We show that for all digraphs D without directed -cycles, χ (D) 4 (D) o( (D)). Because χ 1 (D) = χ A (D), we obtain as a corollary that χ A (D) 4 5 (D) + o( (D)), significantly iproving a bound of Harutyunyan and Mohar. 1. Introduction A proper vertex coloring of an undirected graph partitions its vertices into independent sets. To extend this notion to directed graphs (digraphs), we consider acyclic sets instead of independent sets. An acyclic set in a digraph is a set of vertices whose induced subgraph contains no directed cycle. The acyclic chroatic nuber of a digraph D, denoted χ A (D), is then defined to be the iniu nuber of acyclic sets into which the vertices of D can be partitioned. Many upper bounds on the chroatic nuber of undirected graphs are phrased in ters of (G), the axiu degree of G. To extend this notion to directed graphs, there are a few options which easure the axiu degree. Given a digraph D, (D) is the axiu geoetric ean of the in-degree and the out-degree of a vertex in D, and (D) is the axiu arithetic ean of the in-degree and the out-degree of a vertex in D. Notice that if the out-degrees and in-degrees of all vertices in D are equal, then (D) = (D). The acyclic chroatic nuber χ A (D) is one of any chroatic nubers which have been defined for digraphs. Bokal et al. [3] introduced the circular chroatic nuber of a digraph D, denoted χ c (D), as a generalization of the acyclic chroatic nuber. Let S p denote the circle with perieter p, and for x, y S p, let d(x, y) denote the clockwise distance fro x to y. Then the circular chroatic nuber χ c (D) is the infiu of all positive real nubers p for which there exists a function c : V (D) S p such that for each edge uv of Research Science Institute, Massachusetts Institute of Technology, Cabridge MA. 1

2 D, d(c(u), c(v)) 1. Bokal et al. showed that χ c (D) takes on rational values, and oreover, that χ A (D) 1 < χ c (D) χ A (D). Chen et al. [10] have also considered defining a digraph coloring siply as a coloring of the underlying undirected graph, and proved bounds on this chroatic nuber in ters of the lengths odulo k of directed cycles of the digraph. In this paper, we use the following generalization of the acyclic chroatic nuber and deduce new bounds on the acyclic chroatic nuber itself as a special case of our results. For a positive integer, a digraph D is said to be weakly -degenerate if for every induced subgraph of D, there is a vertex of out-degree or in-degree strictly less than. Therefore, a digraph is weakly 1-degenerate if and only if it is acyclic. Given a positive integer k, a (k, )-degenerate coloring of D is a partition of V (D) into k sets, each of which is weakly -degenerate. Given a positive integer, we denote χ (D), the -degenerate chroatic nuber of D, as the sallest positive integer k such that D has a (k, )-degenerate coloring. Notice that χ 1 (D) = χ A (D); hence the paraeter χ (D) is a generalization of χ A (D). Bokal et al. [3] showed soe further connections between weak degeneracy and acyclic colorings. For instance, if a digraph is weakly -degenerate, then χ A (D) + 1; oreover, this bound is tight for each positive integer. The acyclic chroatic nuber has received the ost attention aong digraph chroatic nubers because recent results [1, 3, 15, 1, ] suggest that the acyclic chroatic nuber in digraphs behaves siilarly to the chroatic nuber in undirected graphs. Much still reains to be learned however. For instance, it is easily proved using the greedy algorith that χ A (D) (D) + 1; this is analogous to the fact that in an undirected graph G, χ(g) (G) + 1. However, this bound is not tight for ost digraphs. In the case of undirected graphs, Brooks [7] ade the first iproveent on the obvious bound of χ(g) (G) + 1; he showed that χ(g) (G) unless G is a coplete graph or an odd cycle. Borodin and Kostochka [6] and Catlin [9] then independently strengthened Brooks theore, showing that if G is K 4 -free, then χ(g) 3( (G)+1). Mohar [18] recently 4 proved an analogue of Brooks theore for digraphs such that at ost one edge connects any pair of vertices; such a digraph is called an oriented graph. It follows fro Mohar s results that if D is an oriented graph, then χ A (D) (D) unless D is a directed cycle. However, it appears that this upper bound on χ A (D) can be significantly iproved further; Harutyunyan and Mohar [1] credit McDiarid and Mohar with the following conjecture. Conjecture ) 1.1 (McDiarid & Mohar, 00 [1]). Every oriented graph D satisfies χ A (D) = O. ( (D) log (D) Conjecture 1.1 is analogous to a result for undirected graphs by Ki [16], who showed that χ(g) (1 + o(1)) (G) if the girth (length of the shortest cycle) of G is greater than log (G) 4. Johansson [14] later extended Ki s bound to graphs G of girth greater than 3, and Jaall [13] has since used a sipler proof to strengthen Johansson s bound by a constant factor. Ki, Johansson, and Jaall all used the pseudo-rando ethod to prove upper bounds on the chroatic nuber. The central idea of this ethod is that given a graph G, and the goal of using (G)/k colors, where k is a positive integer, the algorith goes through several rounds and assigns colors randoly to a subset of the uncolored vertices at

3 each round. Eventually a proper coloring of all the vertices of G will be found with positive probability. Harutyunyan and Mohar [11] applied the pseudo-rando ethod to digraphs to show that χ A (D) (1 e 13 ) (D), which only slightly iproves upon the trivial bound of χ A (D) (D) + 1 and is far fro the bound given in Conjecture 1.1. Therefore, Harutyunyan and Mohar [11] posed the following relaxation of Conjecture 1.1. Conjecture 1. (Harutyunyan & Mohar, 011 [11]). Let D be an oriented graph. Then χ A (D) + 1. (D) In Theore 1.3, we prove an upper bound on the -degenerate chroatic nuber of any digraph D in ters of (D). It is a special case of Theore 1.3 that we can significantly iprove Harutyunyan and Mohar s bound, thus aking progress towards Conjecture 1.. Theore 1.3. Let be a positive integer. Suppose we are given an oriented graph D with (D). Then (D) ( ) 1 (D)+1/ +1/ χ (D) + 1. By taking = 1, the following corollary follows iediately fro Theore 1.3. Corollary 1.4. For an oriented graph D with (D), χ A (D) 4 5 (D) We prove Theore 1.3 by using a strategy siilar to one originally introduced by Borodin and Kostochka [5] and by Catlin [9] to prove an upper bound on the chroatic nuber in undirected graphs. Specifically, we show that the vertices of a directed graph D can be partitioned into several subsets, each inducing a subgraph D i D, such that (D i ) is sall. We then use this to show that χ (D i ) is also sall, eaning that we can find an upper bound on χ (D) i χ (D i ). The organization of this paper is as follows. In Section, we prove Theore 1.3, giving an upper bound on χ (D) for any digraph D in ters of (D). In Section 3, we iprove upon this bound for = 1 for a particular class of digraphs. We give soe concluding rearks in the final section.. Acyclic colorings Recall that Harutyunyan and Mohar [11] proved that given a digraph D, if (D) is large enough, then χ A (D) (1 e 13 ) (D). They used a non-constructive ethod to do so, and posed the proble of iproving this bound, rearking that a different technique ay be necessary. In this section, we use a constructive technique to prove Theore 1.3, a specific case of which (Corollary 1.4) is the significantly stronger upper bound of χ A (D) 3

4 4 (D) + o( (D)). The outline of our proof of Theore 1.3 is soewhat siilar to that of 5 an undirected analogue proved independently by Borodin and Kostochka [6] and Catlin [9]. It is easy to show that χ (D) (D) + 1; the proof is siilar to that of the fact that χ A (D) (D) + 1. In particular, we color the vertices of D greedily, in any order. At each step, the next vertex v to be colored has either out-degree or in-degree at ost (D), suppose without loss of generality out-degree. Therefore, there are at ost (D) colors which are represented in at least out-neighbors of v. We now color v using one of the reaining colors which is represented in less than out-neighbors of v. The resulting coloring is indeed -degenerate, since in any subset of any color class, the vertex in that subset colored last ust have less than in-neighbors or out-neighbors in that subset. Note that Theore 1.3 gives a significant iproveent to χ (D) (D) + 1 for any oriented graph D. To prove Theore 1.3, we begin by generalizing a directed graph analogue of Brooks theore [7] due to Mohar [18] to the fraework of -degenerate colorings. Our proofs follow siilar outlines to those of Mohar. A few definitions are needed to state Lea.1. Given a digraph D and u V (D), we denote the subgraph induced on V (D)\{u} by D u. Moreover, given a positive integer, a critical vertex is a vertex v V (D) such that χ (D v) < χ (D). If every vertex of D is critical and χ (D) = k, then we define D to be a (k, )-critical digraph. Lea.1 shows that critical vertices in a digraph ust have large in-degree and out-degree. Lea.1. Suppose v is a critical vertex in a digraph D, 1, and χ (D) = k. Then d + (v), d (v) (k 1). Proof. Suppose for the purpose of contradiction that d + (v) < (k 1). We will show that we can find a (k 1, )-degenerate coloring of D, a contradiction to the fact that χ (D) = k. Since v is (k, )-critical, we can find a (k 1, )-degenerate coloring of D v. At least one color class c ust be represented in less than out-neighbors of v because otherwise v would have at least (k 1) out-neighbors. Now we color v color c, and clai that the subgraph H induced by all vertices of color c is -degenerate. To see this, let H be an induced subgraph of H. If v V (H ), then notice that v has at ost 1 out-neighbors in H. Otherwise, note that H is a subset of a color class in a (k 1, )-degenerate coloring of D v, eaning that there is soe vertex in H of in-degree or out-degree less than. A siilar arguent shows that d (v) (k 1). A digraph is weakly connected if the underlying undirected graph is connected. We next state and prove Lea., which states that we only need to consider the weakly connected coponents of a digraph to find its -degenerate chroatic nuber. Lea.. If D is a digraph and D 1,..., D l are its weakly connected coponents for soe positive integer l, then for any 1, χ (D) = ax 1 i l χ (D i ). Proof. Let k = ax 1 i l χ (D i ). We can find a (k, )-degenerate coloring of each D i, for 1 i l, and the resulting coposite coloring is a (k, )-degenerate coloring of D since 4

5 there is no edge between any two weakly connected coponents of D. By Lea., a digraph D which is (k, )-critical is also weakly connected. Lea.3 shows that in a (k, )-critical oriented graph, we can find a certain set of vertices, which, when reoved, does not break the weakly connectedness of the oriented graph. Lea.3. Suppose 1, k 3, and D is a (k, )-critical oriented graph on n vertices in which each vertex v satisfies d + (v) = d (v) = (k 1). Then there exists a set of vertices u 1, u,..., u +1, u n V (D) such that u 1,..., u +1 are all out-neighbors or all in-neighbors of u n and the digraph induced by V (D) {u 1,..., u +1 } is weakly connected. Proof. Pick any + 1 vertices u 1,..., u +1 which are all in-neighbors or out-neighbors of another vertex u n, and let u +,..., u n 1 be the reaining vertices of D. Let D 0 be the subgraph induced by {u +,..., u n }. Assue that D 0 is not weakly connected; if it is, the proof is coplete. Let w be any vertex in D 0. Since D is weakly connected, there is soe path (not necessarily directed) fro w to u i, for each 1 i + 1. Therefore, any weakly connected coponent of D 0 ust contain a vertex which is adjacent to soe u i, for 1 i + 1. Given a weakly connected coponent C of D 0, let s(c) be the set of all u i (1 i + 1) which are adjacent to soe vertex of C, and let N(C) be the subgraph of D induced by V (C) s(c). We now perfor a ulti-step process, considering any possible choices of u 1,..., u +1, u n until we find one which forces D 0 to be weakly connected. If D 0 is not weakly connected, then choose a weakly connected coponent C 1, and suppose without loss of generality that s(c 1 ) = {u 1,..., u h } for soe 1 h + 1. We first ark all vertices of C 1. Let C be another weakly connected coponent of D 0. We next clai that there is soe vertex v V (C ) with at least + 1 in-neighbors or + 1 out-neighbors in V (C ). To show this, we ust consider two cases: Case 1. or k 4. Notice that each v V (C ) has (k 1) in-neighbors and (k 1) out-neighbors, all of which ust belong to V (N(C )) by assuption. Since and k 4, we have that (k 4) > + 1 s(c ). Therefore, since D is oriented, each v V (C ) ust have strictly fewer than (k ) in-neighbors or strictly fewer than (k ) out-neighbors in s(c ). Thus v has at least + 1 in-neighbors or + 1 out-neighbors in V (C ). Case. = 1 and k = 3. In this case, we consider acyclic colorings, and oreover, since each v V (C ) has in-neighbors and out-neighbors, our clai is true unless s(c ) = and each v V (C ) has 1 in-neighbor and 1 out-neighbor in s(c ). Moreover, soe vertex of s(c ) ust have an in-neighbor or out-neighbor which does not belong to V (N(C )); if this were not true, then D would not be weakly connected. Since this vertex has a total of 4 neighbors and ust be joined to each vertex of C, we have V (C ) 3. We first assue that V (C ) = 3, and let s(c ) = {w 1, w }. Since each vertex of D has in-neighbors and out-neighbors, and since w 1 and w are interchangeable, N(C ) is 5

6 (a) w (b) w (c) w x w 1 w 1 w 1 Figure 1: (a) The uncolored digraph N(C ) when V (C ) = 3. (b) An acyclic -coloring of N(C ) when w 1 and w are colored the sae color. (c) An acyclic -coloring of N(C ) when w 1 and w are colored different colors. isoorphic to the digraph shown in Figure 1(a). Moreover, since D is (3, 1)-critical, we can find an acyclic -coloring of D V (C ). If w 1 and w have the sae color, suppose red, in this coloring, then suppose without loss of generality that there is no onochroatic directed path fro w 1 to w in D V (C ). We can ake this assuption because the red vertices for an induced acyclic subgraph and since w 1 and w are interchangeable. Next, pick a vertex x V (C ) which has w 1 as its out-neighbor and w as its in-neighbor; there exists such an x since V (C ) = 3 and at ost vertices in V (C ) have w 1 as an in-neighbor and w as an out-neighbor. Color vertex x red, and color the other vertices in V (C ) the other color, suppose blue. As shown in Figure 1(b), this copletes an acyclic -coloring of D, contradicting its (3, 1)-criticality. If w 1 and w have different colors in the acyclic -coloring of D V (C ), then no onochroatic directed cycle in D can contain a vertex in V (C ) and a vertex in D V (N(C )). Assuing that w 1 is colored red and w is colored blue, we pick vertices of V (C ), which, when colored red, do not for a onochroatic directed cycle with w 1, and color these vertices red. We color the third vertex of V (C ) blue. As shown in Figure 1(c), this copletes the acyclic -coloring of D in this case also, contradicting the (3, 1)-criticality of D. In the case that C has or fewer vertices, the proof that we can find an acyclic -coloring of D is nearly identical. Hence we can find a v V (C ) with + 1 in-neighbors or + 1 out-neighbors in V (C ). We let u n = v, and let the +1 in-neighbors or out-neighbors of v in V (C ) be u 1,..., u +1. We finally ark all of u 1,..., u h ; notice that the subgraph induced by the arked vertices, naely N(C 1 ), is weakly connected. We next repeat the above process with our choice of u 1,..., u +1, u n. Notice that the arked vertices ust all belong to the sae weakly connected coponent of D 0 := D {u 1,..., u +1}, which we call C 1. We then find a second weakly connected coponent of D 0, C, and notice that by the sae reasoning as above, there is soe v V (C ) with at least + 1 in-neighbors or + 1 out-neighbors in C. We let u n = v, and let the + 1 in-neighbors or out-neighbors of v in C be u 1,..., u +1. We finally ark all vertices of N(C 1), noting that we ark at least 1 vertex, naely all vertices in s(c 1), for the first tie 6

7 and the subgraph induced by the arked vertices, naely N(C 1), is weakly connected. Since at least 1 new vertex is being arked with each iteration of the process, it ust eventually end at soe step t. That is, we can eventually find u (t) 1,..., u (t) +1, u (t) n so that all vertices in D (t) 0 = D {u (t) 1,..., u (t) +1} are arked, eaning that D (t) 0 is weakly connected. This is a contradiction to our original assuption. Theore.4 states that the in-degree and out-degree of every vertex cannot be too sall in a (k, )-critical oriented graph. Theore.4. Suppose that 1 and D is a (k, )-critical oriented graph in which each vertex v satisfies d + (v) = d (v) = (k 1). Then k. Theore.4 is of particular interest since it generalizes the following theore of Mohar, who proved the case = 1, which is a stateent about acyclic colorings. Theore.5 (Mohar [18]). If D is a (k, 1)-critical oriented graph in which each vertex v satisfies d + (v) = d (v) = k 1, then k. We now prove Theore.4, using Leas.3 and.. Proof of Theore.4. We assue that k 3 for the purpose of contradiction and create a linear ordering of the vertices of D, as follows. Pick a u n and choose + 1 of its outneighbors (or in-neighbors) u 1, u,..., u +1, so that the digraph D {u 1,..., u +1 } is weakly connected. This construction is possible by Lea.3. Let D = D {u 1,..., u +1 }. Now, since u n has (k 1) in-neighbors (or out-neighbors), there is soe u n 1 D apart fro u 1, u,... u +1 such that u n is an out-neighbor or in-neighbor of u n 1. Thus, u n 1 has at ost (k 1) 1 out-neighbors (or in-neighbors) in D u n. Now continue in a siilar anner, using the weakly connectedness of D to find {u +,..., u n 1 } such that each of these vertices v i ( + i n 1) has out-degree or in-degree less than (k 1) in D {u i+1,..., u n }. We now color the vertices of D as follows: we give u 1, u,..., u +1 the sae color; at this point, the coloring of {u 1,..., u +1 } is (k 1, )-degenerate since each vertex aong u 1,..., u +1 has less than in-neighbors or out-neighbors. Then for + i n 1, we start with a (k 1, )-degenerate coloring of {u 1,..., u i 1 } and extend this coloring to u i. This is possible since in the subgraph of D induced by {u 1,..., u i }, u i has in-degree or outdegree less than (k 1). Therefore, in the (k 1, )-degenerate coloring of {u 1,..., u i 1 }, one of the color classes contains fewer than in-neighbors or out-neighbors of u i. We finally extend the (k 1, )-degenerate coloring of {u 1,..., u n 1 } to u n. This is possible since u n has + 1 in-neighbors or out-neighbors, naely u 1,..., u +1, of the sae color, but exactly (k 1) in-neighbors and out-neighbors in total, so we can find a color represented aong fewer than in-neighbors or out-neighbors of u n, and color u n this color. At the end of the process, we clai that each color class c is -degenerate. To show this, for any color class c and subset S of the vertices colored c, pick u i S so that i is as large as possible. Then since u i is the vertex in S that was colored last, u i has at ost 1 in-neighbors or out-neighbors in S, copleting the proof. 7

8 Lea.6 uses Lea.1 to extend Theore.4 to digraphs that are not (k, )- critical. Intuitively, this is possible because (k, )-critical digraphs are the worst case for finding an -degenerate coloring with few colors. Lea.6. Suppose that 1, χ (D) = k + 1, for soe integer k, and that D is an oriented graph. Then (D) > k. Proof. Fix 1. Suppose for the purpose of contradiction that for soe k, there is an oriented graph D with as few vertices as possible, such that χ (D) = k + 1 and (D) k. Notice that if D were not (k + 1, )-critical, we could reove soe vertex v to for D = D v, and we would have χ (D ) = k + 1 and (D ) (D) k. This contradicts the fact that D has as few vertices as possible such that (D) k holds. Hence D is (k + 1, )-critical. By Lea.1, for each v V (D), we have that d + (v) k and d (v) k. In order to have (D) k, we ust have d + (v) = d (v) = k for all v V (D). But then by Theore.4, we have that k + 1, contradicting the fact that k The following corollary follows fro Lea.6. It is a directed analogue of a theore of Borodin [4]. Corollary.7. If D is an oriented graph such that (D), then χ (D) (D). Proof. Let k = eaning that (D) (D). Then k. If χ (D) k + 1, then by Lea.6, (D) > k,. > k, so it is ipossible that k = (D) To prove Theore 1.3, we also use a theore of Lovász [17], which states that the vertices of a graph can be decoposed into sets so that the su of the axial degrees of all the sets is less than the axial degree of the graph. Theore.8 (Lovász [17]). For an undirected graph G, suppose that for soe s 1 and positive integers 1,..., s, we have (G) = (s 1) + s i=1 i. Then there is a covering of V (G) with s subgraphs G i (1 i s), so that (G i ) i for 1 i s. We deduce as a corollary of Theore.8 a version for directed graphs. Corollary.9. Given a digraph D, s 1, and rational nubers 1,..., s 1 so that for 1 i s, i is an integer, suppose (D) = s 1 + s i=1 i. Then there is a covering of V (D) with s subgraphs D i (1 i s) such that (D i ) i. Proof. Given a digraph D, consider the underlying undirected graph G. Notice that (G) = (D). By Theore.8, we can partition G into vertex-disjoint subgraphs G 1,..., G s, so that the axiu total degree in G i is at ost i. For 1 i s, we let D i be the directed subgraph of D induced by V (G i ). Thus, for 1 i s, (D i ) i. 8

9 We now prove Theore 1.3, using Lea.6 and Corollary.9 to find an upper bound on χ (D). Proof of Theore 1.3. Set s = (D) + 1/ + 1/, r = (D) + 1/ s( + 1/). Then (D) = s + ( s i=1 ) + (r 1 ), eaning that, by Corollary.9, the vertices of D can be covered with s + 1 subgraphs D 1,..., D s+1, which satisfy: { if 1 i s (D i ) r 1 if i = s + 1. By Lea.6, if χ (D i ) 3, then we would have that (D i ) >. Moreover, recalling the trivial bound that χ (D) + 1, we have We thus have (D) { if 1 i s χ (D i ) 1 + if i = s + 1. (r 1/) χ (D) s+1 χ (D i ) i=1 s + (r 1/)/ / (D) ( + 1/) +1/ +1/ = / ( ) (D) + 1/ + 1/ = / (D) ( ) 1 (D)+1/ +1/ = Better bounds for = 1 with iproved decoposition lea In this section, we show that the bounds in Theore.8 can be iproved for any digraph which does not contain any of the graphs shown in Figure as an induced subgraph. This leads to iproveents to the bounds in Theore 1.3 in the case = 1 for these classes of digraphs. We first introduce soe notation. Given disjoint sets of vertices V 1 and V which 9

10 F 1 F G 1 G Figure : The digraphs F 1, F, G 1, and G. belong to a digraph D, we let E(V 1, V ) be the set of all edges which point fro a vertex in V 1 to a vertex in V. Moreover, we define e(v 1, V ) = E(V 1, V ). Finally, given a vertex u, we define d + V 1 (u) as the nuber of out-neighbors of u in V 1, d V 1 (u) as the nuber of in-neighbors of u in V 1, and d V1 (u) = d+ V (u)+d 1 V (u) 1. We let F 1, F, G 1, and G be the 4-vertex digraphs shown in Figure. For siplicity, we say that D avoids F if D contains neither F 1 nor F as an induced subgraph and that D avoids G is D contains neither G 1 not G as an induced subgraph. Notice that in Theore 1.3 for the case = 1, we set ost of the i to because it is not true that if (D i ) 1, then χ A (D i ) = 1. The following Lea 3.1 shows that if D avoids F and G, then we can partition D into vertex-disjoint subgraphs D 1,..., D s, so that for all i where i = 1, D i does not contain an induced cycle, and therefore χ A (D i ) = 1. Notice that if the shortest directed cycle in D is of length at least 4, then the constraint is relaxed to the condition that D avoids F. The proof is siilar to that of an undirected analogue proved by Catlin [8]. Lea 3.1. Suppose we are given a digraph D, and positive integers s, 1,..., s. If (D) = (s )/ + s i=1 i and D avoids F and G, then there is a partition of V (D) into s subgraphs D i (1 i s) so that χ A (D i ) i. Proof. Given a partition of V (D) into s subsets V 1,..., V s, define f(v 1,..., V ) = 1 V 1 + V + + s V s + 1 i<j s e(v i, V j ) + e(v j, V i ). We let A be the set of all i such that i = 1. Notice that if i A, then any directed cycle in V i ust be both induced and a coponent of the subgraph induced by V i. Following notation of Catlin [8], we call any such directed cycle a Brooks cycle. Now, choose a partition of the vertices of D into subsets V 1,..., V s so that, (i), f(v 1,..., V s ) is axiized, and (ii), the nuber of Brooks cycles is iniized, subject to (i). We first clai that any partition which axiizes f has (D i ) i for all i. Notice that for any u V 1, and for j s, we have f(v 1,..., V s ) f(v 1 u, V,..., V j + u,..., v s ) 0, 10

11 by axiality of f. But f(v 1,..., V s ) f(v 1 u, V,..., V j + u..., v s ) (1) = e(v 1, V j ) + e(v j, V 1 ) (e(v 1 u, V j + u) + e(v j + u, V 1 u)) + 1 j = d+ V j (u) d V 1 (u) + d V j (u) d + V 1 (u) + 1 j = d Vj (u) d V1 (u) + 1 j. () Therefore, for 1 j s, we have By averaging (3) over all choices of j, 1 j s, we obtain d V1 (u) 1 j + d Vj (u). (3) d V1 (u) 1 s j=1 j s + d G (u) s = 1 (D) (s )/ s + d G (u). s But d G (u) (D), so d V1 (u) 1 + s s. Since 1 and d V1 (u) are integers, we ust have that d V1 (u) 1. Since u can be any vertex in V 1, we have (D 1 ) 1. Moreover, we can repeat the above process with V 1 replaced by V i, for i s, and we have that (D i ) i. Therefore, if i, that is, if i A, then by Lea.6, we have that χ A (D i ) (D i ) i. We now clai that since the partition V 1,..., V s iniizes the total nuber of Brooks cycles subject to the fact that f(v 1,..., V s ) is axiized, the total nuber of Brooks cycles is 0. For the purpose of contradiction, suppose there is soe Brooks cycle C 0 V i where i A, and let v 0 V (C 0 ). If d Vj (v 0 ) j + 1 for each j i, then since d Vi (v 0 ) = i = 1, d(v 0 ) i + ( j + 1 ) = (D) + 1 > (D), j i which is ipossible. Hence there is soe j i so that d Vj (v 0 ) j. Thus, noting that d Vi (u) = i, and by equality of (1) with (), oving v 0 fro V i to V j does not decrease f(v 1,..., V s ). Moreover, since oving v 0 reoves a Brooks cycle fro V i, it ust create a Brooks cycle C 1 in V j. Therefore, by our definition of Brooks cycle, j = 1, so j A. We then pick v 1 v 0 such that v 1 V (C 1 ), and repeat the process. Eventually, since the total nuber of vertices is finite, there ust be soe V i so that the process generates an infinite sequence of Brooks cycles in V i, which we call C 1, C,.... Since each of the steps in our process oves only a single vertex, we ust be able to find Brooks cycles C p and C q that differ in only one vertex. Specifically, suppose that a and the vertices of C p are x 1,..., x a, x p while the vertices of C q are x 1,..., x a, x q. 11

12 If a =, then the vertices x 1, x a, x p, and x q for an induced subgraph isoorphic to either G 1 or G, depending on whether there is an edge between x p and x q. If a 3, then since C p and C q are induced, there is no edge between x a and x 1, eaning that the vertices x 1, x a, x p, and x q for an induced subgraph isoorphic to either F 1 or F. In either case, we have a contradiction to the fact that D is F -free and G-free. Thus there are no Brooks cycles, eaning that for i A, D i is acyclic, so χ A (D i ) = 1 = i for all i A. Our ain theore of this section iproves the bound of χ A (D) 4 5 (D) + o( (D)) in Corollary 1.4 to χ A (D) 3 (D) + o( (D) for oriented graphs D which avoid F and G. Theore 3.. Suppose D is an oriented graph which avoids F and G. Then χ A (D) 3 (D) + 1/ + 1. Proof. Set (D) + 1 s =, r = 3/ (D) + 1 3/s. Then (D) = (r 1) + s + s i=1 1, eaning that, by Lea 3.1, the vertices of D can be covered with s + 1 subgraphs D 1,..., D s+1, which satisfy: { 1 if 1 i s χ A (D i ) r if i = s + 1. Thus χ A (D) s+1 χ A (D i ) i=1 s + (D) + 1/ 3/s + 1 (D) + 1 (D) + 1 = + (D) + 1/ 3/ + 1 3/ 3/ (D) + 1 = (D) + 1/ 1/ + 1 3/ /3 (D) + 1/ Concluding rearks In this paper, we have proved an upper bound on a generalization of the acyclic chroatic nuber, the -degenerate chroatic nuber. Moreover, the special case of = 1 gives a bound on the acyclic nuber which significantly iproves current bounds. However, the 1

13 bound in Theore 1.3 differs fro the conjectured bound in Conjectures 1.1 by a factor of log (D). It sees that a new technique is necessary to obtain the additional factor of log (D), if it is indeed correct. Moreover, it sees that Conjecture 1.1 can be extended to the -degenerate chroatic nuber. Conjecture 4.1. If is a positive integer and c is a constant which does not depend on, then every oriented graph D has χ (D) (c + o(1)) ( (D)/) log( (D)/). We finally note that the bounds obtained in this paper were all phrased in ters of (D). By the arithetic ean - geoetric ean inequality, for any digraph D, we have (D) (D), eaning that bounds phrased in ters of (D) would be slightly stronger. It does not see that our ethods can produce bounds which would hold with (D) replaced by (D). However, ost graphs which are interesting for the proble of acyclic chroatic nuber have the in-degree and out-degree of the vertices approxiately equal, eaning that (D) (D). 5. Acknowledgeents I would like to thank David Rolnick for his helpful discussions and review of the paper. I would also like to thank Jacob Fox for suggesting the direction of research and Tanya Khovanova for helpful suggestions and review. I thank Pavel Etingof, David Jerison, and Slava Gerovitch for coordinating the research, and John Rickert for helpful review. Finally I would like to thank the Center for Excellence in Education, the Research Science Institute, and the MIT Math Departent for their support. References [1] Ron Aharoni, Eli Berger, and Ori Kfir. Acyclic systes of representatives and acyclic colorings of digraphs. Journal of Graph Theory, pages , 008. [] Ido Ben-Eliezer, Michael Krivelevich, and Benny Sudakov. The size Rasey nuber of a directed path. Journal of Cobinatorial Theory, Series B, 10: , 01. [3] Drago Bokal, Gasper Fijavž, Martin Juvan, P. Mark Kayll, and Bojan Mohar. The circular chroatic nuber of a digraph. Journal of Graph Theory, pages 7 40, 004. [4] Oleg Borodin. On decoposition of graphs into degenerated subgraphs. Diskretnyj Analiz, 8, [5] Oleg Borodin. On acyclic colorings of planar graphs. Discrete Matheatics, 5: 11 36, [6] Oleg Borodin and Alexandr Kostochka. On an upper bound of a graph s chroatic nuber, depending on the graph s degree and density. Journal of Cobinatorial Theory, Series B, 3: 47 50,

14 [7] Rowland Brooks. On coloring the nodes of a network. Matheatical Proceedings of the Cabridge Philosophical Society, 37: , [8] Paul Catlin. Another bound on the chroatic nuber of a graph. Discrete Matheatics, 4: 1 6, [9] Paul Catlin. A bound on the chroatic nuber of a graph. Discrete Matheatics, : 81 83, [10] Zhibin Chen, Jie Ma, and Wenan Zang. Coloring digraphs with forbidden cycles. arxiv, , 014. [11] Ararat Harutyunyan and Bojan Mohar. Strengthened Brooks theore for digraphs of girth at least three. Electronic Journal of Cobinatorics, 18: P195, 011. [1] Ararat Harutyunyan and Bojan Mohar. Two results on the digraph chroatic nuber. Discrete Matheatics, 31(10): , 01. [13] Mohaad Shoaib Jaall. A Brooks theore for triangle-free graphs. arxiv, , 011. [14] A. Johannson. Asyptotic choice nuber for triangle free graphs. Unpublished. [15] Peter Keevash, Zhentao Li, Bojan Mohar, and Bruce Reed. Digraph girth via chroatic nuber. SIAM Journal on Discrete Matheatics, 7(): , 013. [16] Jeong Han Ki. On Brooks theore for sparse graphs. Cobinatorics, Probability, and Coputing, 4(): 97 13, [17] László Lovász. On decoposition of graphs. Studia Scientiaru Matheaticaru Hungarica, 1: 37 38, [18] Bojan Mohar. Eigenvalues and colorings of digraphs. Linear Algebra and its Applications, 43: 73 77,