On Brooks Coloring Theorem

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1 On Brooks Coloring Theorem Hong-Jian Lai, Xiangwen Li, Gexin Yu Department of Mathematics West Virginia University Morgantown, WV, Abstract Let G be a connected finite simple graph. δ(g), (G) and χ(g) denote the minimum degree, the maximum degree and the chromatic number of G, respectivly. Brooks theorem states χ(g) (G) + 1. Szekeres and Wilf [J. Combinatorial Theory, 4 (1968) 1-33] and Cao [Linear Algebra Appl., 270 (1998) 1-13] extended Brooks Theorem by replacing (G) by a generic function γ(g) satisfying certain conditions. However, little is known about the graphs satisfying χ(g) = γ(g) + 1. In this paper, we extend the results by Szekeres and Wilf and by Cao to group colorings and investigate the structure of graphs satisfying the equality. Former results are obtained as applications. 1 Introduction Graphs in this note are finite and simple. Unless otherewise stated, we follow [1] for notations and terminology in graph theory. Thus δ(g), (G) and χ(g) denote the minimum degree, the maximum degree and the chromatic number of G, respectivly. We use H G to denote the fact that H is an induced subgraph of G. A graph G is uncyclic if G contains only one cycle. The concept of group coloring is first defined in [4], as the dual concept of group connectivity, which is a generalization of nowhere zero flows. For an (additive) Abelian group A, let F (G, A) denote the set of all maps from E(G) to A. A directed graph G is A-colorable if for any f F (G, A), there is a map c : V (G) A such that for every e = uv E(G), where e is directed from u to v, c(u) c(v) f(e). A graph G is A-colorable if under some orientation of G, G is A-colorable. It is known that that G is A-colorable is independent of the choice of the orientation. The group chromatic number of a graph G is defined to 1

2 be the minimum m for which G is A-colorable for any Abelian group A of order m, and is denoted by χ g (G). It follows from the definitions that, for any G, χ(g) χ g (G). The main purpose of this paper is to seek extensions of the well known Brooks coloring theorem, as stated below. Theorem 1.1 (Brooks, [2]) Let G be a connected graph. Then χ(g) (G) + 1, where equality holds if and only if G is complete or an odd cycle. Szekeres and Wilf [7], and Cao [3], respectively considered the following extention of Theorem 1.1. Let γ(g) be a real function on a graph G. Consider the following two properties: (P1) If H is an induced subgraph of G, then γ(h) γ(g); (P2) γ(g) δ(g) with equality if and only if G is regular. Szekeres and Wilf [7], and Gao [3] proved the following result: Theorem 1.2 (Szekeres and Wilf [7], Cao [3]) Let γ(g) be any real function on a graph G with properties (P1) and (P2). Then χ(g) γ(g) + 1. However, when G is connected and χ(g) = γ(g) + 1, G is not necessarily a complete graph or an odd cycle. Consider the following example. For a connected G, we define 0 if G = K 1 1 if G = K 2 γ(g) = 2 if G is unicyclic or a tree and G K 2 (G) otherwise. For disconnected graph G, γ(g) = max{γ(g i ) : G i is a connected component of G}. Then γ(g) satisfies (P1) and (P2). Therefore when G is a connected unicyclic graph with an odd girth, such that G is not a cycle, we have χ(g) = γ(g) + 1, but G is neither a cycle nor a complete graph. In Section 2, we extend the results by Szekeres and Wilf and by Cao to group colorings and investigate the structure of graphs satisfying the equality. In Section 3, we present some applications to show that several former results can be obtained as corollaries of our main results. 2

3 2 Main Results An analogue of Brooks group coloring theorem is proved in [5]. Theorem 2.1 (Lai and Zhang [5]) For any connected simple graph G, χ g (G) (G) + 1, where equality holds if and only if (G) = 2 and G is a cycle; or (G) 3 and G is complete. Define a graph G to be χ g (G)(or χ(g))-semi-critical if χ g (G v) < χ g (G)(or χ(g v) < χ(g)), for every vertex v G with d(v) = δ(g). Lemma 2.2 Let G be a graph and v V (G), and H = G v. If d G (v) < χ g (H), then χ g (G) = χ g (H). Proof. Without loss of generality we assume that G is oriented such that all the edges incident with v are oriented from v. Let N(v, G) = {v 1, v 2,..., v l }, l = d(v). Thus for any f : E(H) A, there is a map c : V (H) A such that c(x) c(y) f(xy) for each directed edge xy E(H). Since l < χ g (H) A, there is an a A l i=1 {c(v i) + f(v i )}. Therefore we extend c to c 1 : V (G) A by c(u) if u V (H) c 1 (u) = a if u = v. It follows that G is A-colorable and hence χ g (G) = χ g (H). Lemma 2.3 If G is χ g (G)-semicritical with χ g (G) = k, then d G (v) k 1 for all v V (G). Proof. By contradiction, assume that G has a vertex v 0 with d G (v 0 ) = δ(g) k 2. Since G is a χ g -semi-critical graph with χ g (G) = k, we have χ g (G v 0 ) k 1. Let A be an abelian group with A k 1. Then G v 0 is A-colorable. Since d G (v 0 ) < k 1, by Lemma 2.2, χ g (G) = χ g (G v 0 ) k 1, contrary to χ g (G) = k. Theorem 2.4 For any connected simple graph G, χ g (G) γ(g) + 1. If G is χ g (G)-semi-critical, then χ g (G) = γ(g) + 1 if and only if G is a cycle or a complete graph. 3

4 Proof. Let k = χ g (G). Let H G be a k-semi-critical induced subgraph. By (P1), γ(h) γ(g). By Lemma 2.3, δ(h) k 1. Thus we have k 1 δ(h) γ(h) γ(g), which implies that χ g (G) = k γ(g) + 1. Now consider the case when the equality holds. By (P2), γ(c) = 2 for any cycle C and γ(k n ) = n 1. Hence χ g (G) = γ(g) + 1 for a cycle or a complete graph. Conversely, let G be a χ g (G)-semi-critical graph such that χ g (G) = γ(g) + 1. By Lemma 2.3 and (P2), χ g (G) 1 δ(g) γ(g) = χ g (G) 1, which implies δ(g) = γ(g) and then G is regular by (P2). Therefore χ g (G) = γ(g) + 1 = δ(g) + 1 = (G) + 1 and by Theorem 2.1, G is a cycle or a complete graph. Corollary 2.5 Let γ(g) be any real function satisfying (P1) and (P2), then χ(g) γ(g) + 1. If G is semi-critical, then χ(g) = γ(g) + 1 if and only if G is an odd cycle or a complete graph. Next we shall study the structure of connected graphs with χ g (G) = γ(g) + 1. For a positive integer m, define F(m) to be the set of connected simple graphs such that: (1) For m = 1, G F(m) if and only if G is a tree and G K 2. (2) For m = 2, G F(m) if and only if G = K 2. (3) For m = 3, G F(m) if and only if either G is a cycle, or H = G v for some H F(m) and a vertex v with d(v) = 1. (4) For m 4, G F(m) if and only if either G = K m, a complete graph of order m, or H = G v for some H F(m) and a vertex v with d(v) m 2. From the definition, we see that if m = 3, G is a set of unicyclic graphs and if m 4, then F(m) is a set of graphs whose only maximum induced complete subgraph is a K m. Now we prove that when χ g (G) = γ(g) + 1, G must be in F(m) with m = χ g (G). Theorem 2.6 For any connected simple graph G, if χ g (G) = γ(g) + 1, then G F(m) with m = χ g (G). Proof. If χ g (G) = 1, 2, it is trivial. 4

5 Let χ g (G) 3 and H 0 be a χ g (G)-semi-critical induced subgraph of G, then χ g (H 0 ) = χ g (G) = γ(g) + 1 γ(h 0 ) + 1 χ g (H 0 ). So all equalities hold, we have χ g (H 0 ) = γ(h 0 ) + 1 and γ(h 0 ) = γ(g). Thus H 0 is a cycle or a complete graph by Theorem 2.4. Let m = χ g (H 0 ) = χ g (G), and then δ(h 0 ) = m 1. If χ g (G) = γ(g) + 1 and δ(g) m 1, then m = χ g (G) = γ(g) + 1 δ(g) + 1 m, and so γ(g) = δ(g) = m 1. Thus G is regular and since G is connected, G = H 0. For m = 3, H 0 is a cycle and so G F(3); for m 4, H 0 is a complete graph and so G F(m). Now we assume that χ g (G) = γ(g) + 1 and δ(g) m 2. Note that G cannot be a cycle (when m = 3) or a complete graph (when m 4). By Theorem 2.4, G cannot be semicritical. Therefore, there exists v V (G) with d(v) = δ(g) such that χ g (G) = χ g (G v). We use induction on V (G) to prove that G F(m). If V (G) = V (H 0 ) + 1, then G v = H 0 F(m), and by the definition of F(m), G F(m). Therefore we can assume that V (G) > V (H 0 ) + 1. Then by Lemma 2.2 and Property (P1), χ g (G v) = χ g (G) = γ(g) + 1 γ(g v) + 1 χ g (G v). So χ g (G v) = γ(g v) + 1 and by induction hypothesis, G v F(M). Therefore by the definition of F(m), G F(m) with m = χ g (G). Since χ(g) χ g (G) γ(g) + 1, that χ(g) = γ(g) + 1 implies χ g (G) = γ(g) + 1. Therefore when χ(g) = γ(g) + 1, we also have G F(m). Of course, if G is a unicyclic graph, the cylcle can not be even. Thus we have the following corollary, Corollary 2.7 For a connected graph G, if χ(g) = γ(g) + 1, then G F(m) for some m 3 or G is a unicyclic graph with odd girth or G = K 2. 3 Applications of the main results By Theorem 2.6 or Corollary 2.7, once more information about γ(g) is known, the connected graphs satisfying equality in Theorem 2.6 or Corollary 2.7 can be determined. This allows us to apply our main results and to obtain former results as corollaries, as seen below. The eigenvalues of G are the eigenvalues of its adjacenty matrix A(G). The largest eigenvalue of A(G) is called the spectral radius of G, denoted by ρ(g). Here are some results with respect to ρ(g). 5

6 Lemma 3.1 Each of the following holds: (i) ρ(k n ) = n 1, ρ(c n ) = 2. (ii) ρ(g) ρ(g v) for any vertex v G. (iii) For a graph G, either 2 E(G) V (G) < ρ(g) < (G) or 2 E(G) V (G) = ρ(g) = (G) and G is regular. The proof of these can be found in Liu and Lai [6], as Exercise 1.7, Theorem and Theorem 1.3.2A. Corollary 3.2 (Wilf [8]) Let G be a connected simple graph with spectral radius ρ(g). Then cycle; (i) χ g (G) ρ(g) + 1, where equality holds if and only if G is a complete graph or a (ii) χ(g) ρ(g) + 1, where equality holds if and only if G is a complete graph or an odd cycle. Proof. By Lemma 3.1, ρ(g) satisfies (P1) and (P2). Thus χ g (G) ρ(g) + 1. If G = K n or G = C, it is easy to see χ g (G) = ρ(g) + 1. Now let χ g (G) = ρ(g) + 1. By Theorem 2.6, G F(m). For m = 1, 2, it is easy to see the result holds. For m = 3, G F(m) implies χ g (G) = 3 and then ρ(g) = 2. As for any G F(3), E(G) = V (G), we have ρ(g) = 2 = 2 E(G) V (G) and so by Lemma 3.1(iii), G is regular. Thus G must be a cycle. For m 4, that G F(m) implies χ g (G) = m and K m G. Thus ρ(g) = m 1. As E(G) m(m 1) 2 + (m 2)(n m), where n = V (G), we have 0 = ρ(g) (m 1) 2 E(G) (m 1) n m(m 1) 2( 2 + (m 2)(n m)) (m 1) = (m 3)(1 m n n ) 0. Therefore all equalities must hold. Thus m = 3 or m = n, ρ(g) = 2 E(G) n regular. So m = n and G = K m. The proof for χ(g) is similar. and G is For a graph G with V (G) = {v 1, v 2,..., v n }, define the average degree of G to be l(g) = 2 E(G) V (G), and the maximum average degree of G to be L(G) = max{l(h) : 6

7 H is an induced subgraph of G}. It is easy to see that L(G) satisfies (P1) and (P2), and so we have Corollary 3.3 Let G be a connected simple graph with maximum average degree L(G). Then (i) χ g (G) L(G) + 1, where equality holds if and only if G = K 1 or G = K 2 or G is unicylic, or for m = χ g (G) 4, G F(m) and L(G) = m 1; (ii) χ(g) L(G) + 1, where equality holds if and only if G = K 1 or G = K 2, or G is a unicyclic graph with an odd girth, or for m = χ g (G) 4 G F(m) and L(G) = m 1. Proof. The proof is routine. Let the 2-degree of vertex v in a graph G, denoted by t(v), be the sum of degrees of the vertices adjacent to v in G, and let T (G) denote the maximum 2-degree of G. Corollary 3.4 (Cao, Theorem 3 of [3]) Let G be a connected simple graph with maximum 2-degree T (G). Then cycle. (i) χ g (G) T (G) + 1, where equality holds if and only if G is complete or a cycle; (ii) χ(g) T (G) + 1, where equality holds if and only if G is complete or an odd Proof. It is easy to check that T (G) satisfies (P1) and (P2). And so χ g (G) T (G)+1, and when χ g (G) = T (G) + 1, G F(m). For m = 1, 2, it is easy to see the result holds. For m = 3, G is a unicyclic graph. Then χ g (G) = 3. Since T (G) x with T (G) = 2 if and only if x = 0, that is, G must be a cycle. Similarly, for m 4, G F(m) and T (G) = m 1 if and only if G is a complete graph. The proof for χ(g) is similar. Define the k-degree of a vertex v in a graph G to be the number of walks of length 1 from v in G. The 2-degree is exactly the same as we defined before. Let k (G) be the maximum k-degree in a graph G which is the maximum row sum of A k, where A is the incident matrix of G. It is not hard to prove that the function [ k (G)] 1 k (P2). Similar to the proof of Corollary 3.4, we have satisfies (P1) and Corollary 3.5 (Cao [3]) Let G be a connected graph with maximum k-degree k (G). Then cycle. (i) χ g (G) [ k (G)] 1 k + 1, where equality holds if and only if G is complete or a cycle; (ii) χ(g) [ k (G)] 1 k + 1, where equality holds if and only if G is complete or an odd 7

8 References [1] J. A. Bondy and U. S. R. Murty, Graph Theory with Applications, American Elsevier, New York, [2] R. Brooks, On colouring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941), [3] D. Cao, Bounds on eigenvalues and chromatic number, Linear Algebra Appl., 270 (1998), [4] F. Jaeger, N. Linial, C. Payan, and M. Tarsi, Graph Connectivity of Graphs A Nonhomogeneous Analogue of Nowhere-Zero Flow Properties, J. Combin. Theory Series B 56 (1992), [5] H.-J. Lai and X. Zhang, Group colorability of graphs, Ars Combinatorics, accepted. [6] B. Liu and H.-J. Lai, Matrices in combinatorics and graph theory, Kluwer Academic Publishers, Dordrecht, [7] G. Szekeres and H. S. Wilf, An inequality for the chromatic number of a graph, J. Combinatorial Theory 4 (1968) 1 3. [8] H. S. Wilf, The eigenvalues of a graph and its chromatic number, J. London Math. Soc. 42 (1967),