University of Twente. Faculty of Mathematical Sciences. A short proof of a conjecture on the T r -choice number of even cycles

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1 Faculty of Mathematical Sciences University of Twente University for Technical and Social Sciences P.O. Box AE Enschede The Netherlands Phone: Fax: Memorandum No A short proof of a conjecture on the T r -choice number of even cycles R.A. Sitters December 1998 ISSN

2 A short proof of a conjecture on the T r -choice number of even cycles R.A. Sitters Faculty of Mathematical Sciences University of Twente P.O. Box AE Enschede, The Netherlands Abstract In this note we prove that the T r -choice number of the cycle C 2n is equal to the T r -choice ( number ) of the path (tree) on 4n 1 vertices, i.e. T r - 4n 2 ch(c 2n )= 4n 1 (2r +2) + 1. This solves a recent conjecture of Alon and Zaks. Key words: graph coloring, list coloring, choosability, choice number, T r -choice number, even cycle. AMS Subject Classifications (1991): 05C15. 1 Introduction We consider simple finite undirected graphs only, and refer to [2] for any undefined terminology and notation. We refer to [1], [3], [4] and [5] for more background information on the coloring problems considered here. Let G =(V,E) be a graph, and let T r = {0, 1,...,r} for some integer r 0. A list assignment L of G is a function that assigns to every vertex v V a nonempty list L(v) of positive integers, i.e. L(v) N for all v V. Given a list assignment L of G, G is called L-list T r -colorable if there is a coloring f : V N such that f(v) L(v) for all v V and f(u) f(v) T r for all u, v V with uv E; inthatcasef is called a feasible coloring (with respect to L and T r ). G is called T r -k-choosable if G is L-list T r -colorable for any list assignment L of G with L(v) = k for all v V.TheT r -choice number of G, denoted by T r -ch(g), is the smallest k for which G is T r -k-choosable. Exact values of the T r -choice number are known for complete graphs, trees, odd cycles, and for the four-cycle C 4 (See [1], [3] and [5] for the details). In [1] Alon and Zaks show that T r -ch(c 2n ) (2r + 2)(4n 2)/(4n 1) +1. They conjecture ([1], Conjecture 3.7) that in fact equality holds. In the next section we give a short proof of their conjecture. 1

3 2 The T r -choice number of C 2n Before we determine the T r -choice number of C 2n we introduce some auxiliary terminology and notation. Suppose P = v 1 v 2...v p is a path with list assignment L. We denote by P (v i,v j )(forsomei, j with 1 i j p) the subpath of P from v i to v j. We denote L i = L(v i ) and define D i and L i for all i {1,...,p} as follows: D 1 =, L 1 = L 1. Given D t and L t for some t with 1 t<p, we define L t+1 = L t+1 \ D t+1, where D t+1 is given by L t+1 if L t = D t+1 = if L t and max(l t) min(l t) > 2r [max(l t) r, min(l t)+r] if L t and max(l t) min(l t) 2r. Clearly D t+1 only depends on L t,andifd i for some i {2,...,p}, thend i is a set of at most 2r + 1 consecutive integers. It is obvious that the elements of D i cannot occur in a feasible coloring of P with respect to L and T r.infact, P (v 1,v i ) has a feasible coloring if and only if L i, andinthatcaseforany feasible coloring f of P (v 1,v i ), f(v j ) L j for all j {1,...,i}. We prove the following positive result on feasible colorings of P, that will be crucial in our proof of the conjecture on T r -ch(c 2n ). Lemma 1. Let P = v 1 v 2...v 2n+1 be a path with list assignment L, andlet D i,l i (i = 1,...,2n + 1) be defined as above. If max(l 1 ) L 2n+1 and min(l 1 ) L 2n+1,thenP has a feasible coloring f such that f(v 1)=f(v 2n+1 ) {min(l 1 ), max(l 1 )}. Proof. Given P, L, D i,l i as in the lemma, suppose max(l 1) L 2n+2 and min(l 1 ) L 2n+1. The idea is to construct a feasible coloring f with the given properties by starting at v 2n+1 and working ourselves backward through P. During this construction we try to alternate between the extreme values of L for the choice of f at the vertices of P. Since L 2n+1, there is a feasible coloring of P (v 1,v 2n+1 ). We use the obvious fact that given a feasible coloring f of P (v i+1,v 2n+1 )withf(v i+1 ) L i+1 for some i 1, this f can be extended to a feasible coloring of P (v i,v 2n+1 )withf(v i ) {min(l i ), max(l i )}. Moreover, if f(v i+1 )=min(l i+1 ) while the extension with f(v i)=max(l i )isnot feasible, then both f(v i )=min(l i ), f(v i+1) =min(l i+1 )andf(v i)=min(l i ), f(v i+1 )=max(l i+1 ) would yield a feasible extension if the respective values of f(v i+1 ) were feasible. A similar statement holds with max and min interchanged at all places. Now start at v 2n+1 with f(v 2n+1 )=min(l 1 )and try to extend f to a feasible coloring of P by adding one by one the predecessors of v 2n+1 on P, and choosing an f-value from {min(l ), max(l )} alternating between the extremes of L for the choice of f if possible. If in this way we reach v 1 with f(v 1 ) = min(l 1 ), then f is a feasible coloring satisfying the conditions of the lemma. If not, then f(v 1 ) = max(l 1 ) and for some 1 j 2n either (f(v j ),f(v j+1 )) = (min(l j ), min(l j+1 )) or (f(v j ),f(v j+1 )) = (max(l j ), max(l j+1 )), while (max(l j ), min(l j+1 )) or (min(l j ), max(l j+1 )) were not feasible, respectively. Suppose that we are 2

4 in the first case. The arguments for the other case are similar and left to the reader. Since (f(v j ),f(v j+1 )) = (max(l j ), min(l j+1 )) is not a feasible extension, the above observations yield that both (min(l j ), min(l j+1 )) and (min(l j ), max(l j+1 )) would have been feasible. Now consider a feasible coloring f of P (v j+1,v 2n+1 )withf (v 2n+1 )=max(l 1 ) obtained by a similar procedure, using only extreme values of L. Then clearly f defined by f (v i )=f(v i ) for 1 i j and f (v i )=f (v i )forj +1 i 2n + 1 is a feasible coloring of P with f (v 1 )=f (v 2n+1 )=max(l 1 ). This completes the proof of Lemma 1. We will apply Lemma 1 to several paths obtained from an even cycle C = w 1 w 2...w 2n w 1 with list assignment L in the following way. A C-path of length k is a path v i v i+1...v i+k for some i {1,...,2n} with list assignment L such that L(v j )=L(w j mod 2n ). Note that k can be any nonnegative integer. In the typical cases below k will exceed the length of C. The main result of this note is the following necessary and sufficient condition for the existence of a feasible coloring of C 2n, yielding an upper bound for T r -ch(c 2n ) that equals the lower bound given in [1]. Theorem 2. Let C = w 1 w 2...w 2n w 1 be an even cycle on 2n vertices with list assignment L. Then C has a feasible coloring if and only if every C-path of length 4n 2 has a feasible coloring. Proof. Let C = w 1 w 2...w 2n w 1 be an even cycle on 2n vertices with list assignment L. IfC has a feasible coloring, then clearly every C-path of length 4n 2 has a feasible coloring. For the converse, assume that every C-path of length 4n 2 has a feasible coloring. Consider the C-path P = v 1 v 2...v 6n with list assignment L(v i )=L(w i mod 2n )(1 i 6n), and define D i,l i as before. We start with some easy observations, given without proofs. (1) D i D i+2n and L i L i+2n (1 i 4n). (2) Each vertex v i (1 i 6n) is of exactly one of the following types: Type 1 if max(l(v i )) D i and min(l(v i )) D i ; Type 2 if max(l(v i )) D i and min(l(v i )) D i ; Type 3 if max(l(v i )) D i and min(l(v i )) D i ; Type 4 if max(l(v i )) D i and min(l(v i )) D i. (3) If v i is of type 1 or type 2 (1 i 4n), then v i+2n is of type 1 or type 2, respectively, unless it is of type 4. (4) If v i is of type 3 (1 i 6n 1) and D i+1, thend i+1 =[max(l(v i )) r, min(l(v i )) + r], implying that D j for the path P (v 1,v j ) is the same as D j for the path P (v i,v j )(i +1 j 6n). (5) If v i is of type 1 (respectively type 2) (1 i 6n 1) and D i+1, then max(d i+1 ) > max(d i )+r (respectively min(d i+1 ) < min(d i ) r). 3

5 We distinguish two cases. Case 1. There is a vertex v i+2n {v 2n+1,...,v 6n } of type 3. Then by (1), v i is of type 3, and Lemma 1 applied to P (v i,v i+2n ) yields a feasible coloring of C. Case 2. There is no vertex in {v 2n+1,...,v 6n } of type 3. Let j be the maximum index such that v j {v 1,...,v 2n } is of type 3. Note that v 1 is of type 3. By assumption, the path P (v j,v j+4n 2 ) has a feasible coloring, so using (4) no vertex of P (v j,v j+4n 2 )isoftype4. Thenv i is of type 1 or type 2 (j +1 i j +4n 2). For convenience we assume j = 1. We next show that L 4n = L 2n. Suppose to the contrary that L 4n L 2n. Then also L 2n+i L i (1 i 2n), since B i and L i only depend on L i 1. We now assume that v 2n+1 is of type 1. The other case (v 2n+1 is of type 2) can be treated analogously. Since v 2n+1 is of type 1, min(d 2n+2 )=max(l 2n+1 ) r =max(l(v 2n+1)) r =min(d 2 ). By (3), v 2n+2 and v 2 are either both of type 1 or both of type 2. Together with L 2n+2 L 2 these observations yield that v 2n+2 and v 2 are both of type 1. For similar reasons v 2n+3 and v 3 are both of type 1, and so on, up to v 4n 1 and v 2n 1. Similarly, v 2n is of type 1, and v 4n is of type 1 or type 4. We conclude that there exists a set of more than 2n consecutive vertices of type 1, for example {v 2,v 3,...,v 2n+2 }.By(1),D 2 D 2n+2, and by (5), max(d 2n+2 ) max(d 2 )+2n(r + 1), hence D 2n+2 > 2n(r +1) > 2r + 1, a contradiction. This proves that L 4n = L 2n. Now Lemma 1 applied to P (v 2n,v 4n ) with assigned lists L 2n,L(v 1),...,L(v 2n ) yields a feasible coloring of C. Theorem 2 clearly implies that C 2n has a feasible coloring if all assigned lists have cardinality k T r -ch(p 4n 1 ). Together with the lower bound given in [1] this proves the following result. Corollary 3. T r -ch(c 2n )= (2r + 2)(4n 2)/(4n 1) +1. Acknowledgment The author thanks H.J. Broersma for his help in the preparation of this paper. References [1] N. Alon and A. Zaks, T -choosability in graphs. Discrete Applied Math. 82 (1998) [2] J. A. BondyandU. S. R. Murty, Graph Theory with Applications. Macmillan, London and Elsevier, New York (1976). [3] B. A. Tesman, List T -colorings of graphs. Discrete Applied Math. 45 (1993) [4] Z. Tuza, Graph colorings with local constraints a survey. Discussiones Math.: Graph Theory 17 (1997)

6 [5] A. O. Waller, Some results on list T -colourings. Discrete Math. 174 (1997)