Cycles and new bounds for the chromatic number

In 1976, Bondy proved that the chromatic number of a graph G is at most its circumference, the length of a longest cycle. In 1992, Tuza proved that, for any integer k > 2, if a graph G contains no cycle of length congruent to 1 modulo k, then chi (G) <= k. In this paper, we use a DFS tree of a connected graph G to give proper colorings of V (G), based on the non-tree edges of the corresponding DFS tree and the cycle lengths of G. This method allows us to obtain improvements of the bounds proposed by Bondy and Tuza, respectively. Moreover, the hypotheses of most theorems presented in this paper can be checked in polynomial time. (c) 2022 Elsevier B.V. All rights reserved.