The colour theorems of Brooks and Gallai extended

One of the basic results in graph colouring is Brooks' theorem [-4] which asserts that the chromatic number of every connected graph, that is not a complete graph or an odd cycle, does not exceed its maximum degree. As an extension of this result, Gallai [6] characterized the subgraphs of k-colour-critical graphs induced by the set of all vertices of degree k 1. The choosability version of Brooks' theorem was proved, independently, by Vizing [9] and by ErdiSs et al. [5]. As Thomassen pointed out in his talk at the Graph Theory Conference held at Oberwolfach, July 1994, one can also prove a choosability version of Gallai's result. All these theorems can be easily derived from a result of Borodin [2, 3] and Erd6s et al. [-5] which enables a characterization of connected graphs G admitting a color scheme L such that IL(x)l ~> d~(x) for all x ~ V(G) and there is no L-colouring of G. In this note, we use a reduction idea in order to give a new short proof of this result and to extend it to hypergraphs. A hypergraph G = (V, E) consists of a finite set V = V(G) of vertices and a set E = E(G) of subsets of V, called edges, each having cardinality at least two. An edge e with I el >/3 is called a hyperedge and an edge e with l el = 2 is called an ordinary edge. The degree d~(x) of a vertex x in G is the number of the edges in G containing x. Let A (G) and 6(G) denote the maximum degree and the minimum degree of G, respectively. If A(G) = 6(G) = r, then G is cal led r-regular. Let X ~_ V(G). The s u b h y p e r g r a p h GEX] of G induced by X is defined as follows: V ( G [ X ] ) = X and E ( G [ X ] ) = {e ~ E(G)Ie ~_ X}; further, G X = G[V(G) X] . F o r x ~ V(G), let G \ x deno te the s u b h y p e r g r a p h of G with V ( G \ x ) = V ( G ) {x} and E ( G \ x ) = E ( G {x})w {e {x}[x e e eE(G) & lel >~ 3}. F o r a hyperedge e, let ( e ) denote the hype rg ra ph (e, {e}). * Corresponding author. 1 Partially supported by the International Science Foundation, grant RPY 300. 0012-365X/96/$15.00 © 1996 Elsevier Science B.V. All rights reserved SSDI 0 0 1 2 3 6 5 X ( 9 5 ) 0 0 2 9 4 4 300 A. V. Kostochka et al. / Discrete Mathematics' 162 (1996) 299-303 Let G be a connected hypergraph. A vertex x of G is called a separating vertex of G iff G \ x is disconnected. An edge e of G is called a bridge of G iff G {e} = (V(G), E ( G ) {e}) has precisely lel components. By a block of G we mean a maximal connected subhypergraph B of G such that no vertex of B is a separating vertex of B. Any two blocks of G have at most one vertex in common and, obviously, a vertex of G is a separating vertex of G iff it is contained in more than one block of G. An end-block of G is a block that contains at most one separating vertex of G. By a brick we mean a hypergraph of the form ( e ) for some hyperedge e, or an odd cycle consisting only of ordinary edges, or a complete graph. A connected hypergraph all of whose blocks are bricks is called a Gallai tree; a Gallai forest is a hypergraph whose components are Gallai trees. Consider a hypergraph G and assign to each vertex x of G a set L(x) of colours (positive integers). Such an assignment L of sets to vertices in G is referred to as a colour scheme (or briefly, a list) for G. An L-colourin 9 of G is a mapping f of V(G) into the set of colours such that f ( x ) • L ( x ) for all x •V(G) and I{ f (x) lx •e}l /> 2 for each e •E(G). If G admits an L-colouring, then G is said to be L-colourable. In case of L ( x ) = {1 . . . . ,k} for all x • V(G), we also use the terms k-colouring and k-colourable, respectively. A hypergraph G is called k-choosable iff G is L-colourable for every list L of G satisfying IL(x)l = k for all x • V(G). The chromatic number ;g(G) (choice number )&(G)) of G is the least integer k such that G is k-colourable (k-choosable). The choosability concept was introduced, independently, by Vizing [9] and ErdSs et al. [5]. We call a pair (G, L) consisting of a connected hypergraph G and a list L for G a bad pair of order n iff Zx~V~G)d~(x) = n, [L(x)] ~> de(x) for all x • V(G) and G is not L-colourable. Reduction. Let (G, L) be a bad pair of order n ~> 1, y a non-separating vertex of G, and e • L(y). For the connected hypergraph G' = G \ y define a list L' = L~ by setting L'(x) = L(x) {c~} if {x, y} • E(G) and L'(x) = L(x) otherwise. Then, clearly, IL'(x)l ~> dw(x) for all x • V(G') and (G', L') is a bad pair of some order k with k < n. Lemma 1. Let (G, L) be a bad pair of order n >t0. Then the following statements hold. (1) ]L(x)[ = d~(x) for all x • V(G). (2) Every hyperedge e of G is a bridge of G and, therefore, ( e ) is a block of G. (3) I f G has no separating vertex, then L(x) is the same for all x • V(G) and G is regular. (4) G is a Gallai tree. Proof(By induction on n). If(G, L) is a bad pair of order n = 0, then L(x) = 0 where x is the only vertex of G and the statements (1)-(4) are obviously true. In what follows, let (G, L) be a bad pair of order n i> 1. A. [d Kostochka et al./Discrete Mathematics 162 (1996) 299-303 301 For the p roof o f ( l ) consider an a rb i t ra ry vertex x of G. Since G is connected, there is a non-separa t ing vertex y ~ x in G. Now, consider the bad pair (G', L') where G' = G \ y and L' = L~ for some ~ eL(y) . Note that L(y) ¢ 0. Then, by the induction hypothesis, [ L' (x) L = dG,(x) which implies immediately that [ L (x)] = d~ (x). This proves (1). To prove (2) suppose that some hyperedge e of G is not a bridge of G. Then, for some vertex x ee , the hyperg raph G ' = (V(G), E ( G ) { e } w [ e ~tx}})is connected and. therefore, (G', L) is a bad pair with ]L(x)L t> de(x) > da,(x), a contradic t ion to (l). This proves (2). Fo r the p roof of (3) assume that G has no separat ing vertex. If G contains a hyperedge e, then, because of (2), G = {e) and (3) is obviously true. Otherwise G is a graph, i.e. G has only ord inary edges, and, for proving (3), it suffices to show that L(x) = L(y) for all x, y e V(G). Suppose that this is not true. Then there are two adjacent vertices x, y in G such that L(x) ¢ L(y). W.l.o.g. assume that ~ e L ( y ) L(xt. Set G ' G \ y = G { y } and L ' = L~. Then, clearly, (G',L') is a bad pair with ]L'(x)L > dG,(x), a contradic t ion to (1). This proves (3). For the p roo f of (4), we consider two possible cases. Case 1: G has a separa t ing vertex. Then G has at least two end-blocks, say B1 and B2. For i = 1, 2, there is a non-separa t ing vertex Yi of G conta ined in Bi. Then, by induct ion (using the reduct ion operat ion), G \ y i is a Gal lai tree and, clearly, every block B va Bi of G is a block of G \ y i , too. This implies that G is a Gal lai tree. Case 2: G has no separa t ing vertex, i.e. G is a block. Because of (2) and (3), we may assume that G is a g raph which is regular of some degree r ~> 1 and L(x) -C for all x e V ( G ) where C is a set of r colours. Let y be a vertex of G and set G' = G \ y = G {y}. Then, by induction, G' is a Gal lai tree and every block of the graph G' is a complete graph or an odd cycle. If G' consists of a single block, then, clearly, bo th G' and G are comple te graphs. So, let G' have at least two blocks. Since every end-block of G' must be (r 1)-regular, the degree of y in G is at least 2(r 1 t which yields r = 2, and, therefore, the graph G is a cycle. Since G is not L-colourable and the same two colours are available at each vertex of the cycle G, we conclude that G is an odd cycle. This proves (4). [] L e m m a 1 is not quite new. In part icular , its g raph version was proved independently by Borodin [2, 3] and Erd6s et al. I-5]. However , their proofs use different ideas and are longer. Proofs of s ta tements (1) and (3) in the graph version based on a sequential colour ing a rgumen t were given by Vizing [-9] and Lov~sz [-7]. The next result is a simple consequence of L e m m a 1. A different p roof of its graph version has recently been given by T h o m a s s e n [8]. Theorem 2. Let L be an arbitrary list.['or a 9iven hypergraph G. Furthermore, let X be a subset o f V(G) such that G [ X ] is connected and IL(x)[ ~> d6(x) for all x e X . Assume that G X is L*-colourable where L* is the restriction of L to V(G) X. U G is not L-colourable, then G [ X ] is a Gallai tree and JL(x)l = dG(x) J~)r each x e X . 302 A. K Kostoehka et aL / Discrete Mathematics 162 (1996) 299-303 Proof. Consider an arbitrary L*-colouring f of G X and choose for each edge e ~E(G) with e X ¢ O, a vertex v(e)E e X. For the connected hypergraph G' = G IX], define a list L' by L'(x) = L(x) { f (v (e ) ) lx e e e E(G) & e X ~ 0} for each x eX. Then [L'(x)[/> da,(x) for every vertex x of G' and, clearly, every L'-colouring of G' yields an L-colouring of G. Therefore, if G is not L-colourable, then (G', L') is a bad pair and, by Lemma 1, G' is a Gallai tree and [L'(x)[ = dG,(x) for all x e X implying that [L(x)[ = de(x) for all x eX. This proves Theorem 2. [] Let G be a connected hypergraph. Clearly, z(G) ~< )0(G) and Theorem 2 implies that z(G) <~ z,(G) <~ A (G) + 1. If G is a brick, i.e., if [E(G)I -1 or G is a complete graph or an odd cycle, then z(G) = z~(G) = A(G) + 1. Brooks [4] proved that the complete graphs and the odd cycles are the only connected graphs whose chromatic number is larger than their maximum degree. This famous result has a number of different proofs. Some of them are listed in [1]. That a similar result is true also for the choice number, was proved, independently, by Vizing [9] and Erd6s et al. [5]. Theorem 3. I f G is a connected hypergraph that is not a brick, then z(G) <~ zt(G) <~ A(G). Proof. Suppose that G is not L-colourable for some list L for G where [L(x)[ = A(G) for each x ~ V(G). Then, by Theorem 2, G is a Gallai tree which is regular of degree A (G). This implies that G is a brick, i.e., a connected hypergraph consisting of exactly one hyperedge, or an odd cycle consisting only of ordinary edges, or a complete graph. This proves Theorem 3. [] Another consequence of Theorem 2 is the following result stated for the chromatic number as well as for the choice number. Theorem 4. Let ~ stand for Z or Zt. Le t G be a hypergraph with ~(G) >>k for some k >>. 2. Furthermore, let X be a subset o f V(G) such that G [ X ] is connected and dG(X) ~ k -1 for each x ~ X . I f ~(G X ) <~ k 1, then the followin9 statements hold. (1) ~(G) = k, (2) de(x) = k l for each x ~ X , and (3) G [-X] is a Gallai tree. A. V, Kostochka et al. / Discrete Mathematics 162 (1996) 299-303 303 Proof . We prove T h e o r e m 4 only for the choice number . Since )~t(G) >~ k, G is not L -co lou rab l e for some list L for G where IL(y)[ = k 1 for each y ~ V(G). Then LL(x)I 1> dG(x) for all x e X . Let L* = L [G-x. The h y p e r g r a p h G X being L*co lourable , (2) and (3) are consequences of T he o re m 2. Fu r the rmore , T he o re m 2 implies tha t G is L -co lou rab l e for every list L sat isfying IL(y)I = k for each y ~ V(G),