Improved bounds on the generalized acyclic chromatic number

Abstract

An r-acyclic edge chromatic number of a graph G, denoted by ar′r(G), is the minimum number of colors used to produce an edge coloring of the graph such that adjacent edges receive different colors and every cycle C has at least min {|C|, r} colors. We prove that ar′r(G) ≤ (4r + 1)Δ(G), when the girth of the graph G equals to max{50, Δ(G)} and 4 ≤ r ≤ 7. If we relax the restriction of the girth to max {220, Δ(G)}, the upper bound of ar′r(G) is not larger than (2r + 5)Δ(G) with 4 ≤ r ≤ 10.