Congruence of cycle lengths and chromatic number

Abstract

AbstractIn this paper we obtain the following upper bound for the chromatic number of a graph : Let be a connected graph with at least one cycle and an integer, with . If is the length of a longest cycle in whose length is congruent to 1 modulo , and is the length of a shortest cycle in whose length is congruent to 1 modulo , then . This generalizes a theorem by Diwan et al. Moreover, we give a polynomial time algorithm to get a proper coloring with such number of colors. Additionally, we give other upper bounds for the chromatic number of a graph, in terms of its cycle lengths. We also give a method to obtain the minimum , for which a graph satisfies the hypotheses of the following two outstanding theorems: (1) Let be a graph, if for some positive integer with contains no cycle of length 1 modulo , then is ‐colorable, Tuza; (2) Let be a graph, if contains no cycle of length modulo for some integers and with and , then is ‐colorable if and ‐colorable otherwise, Chen et al. So we get the best bound that each one of these two results allows.