Improper Coloring of Sparse Graphs with a Given Girth, II: Constructions

Abstract

A graph G is -colorable if can be partitioned into two sets and so that the maximum degree of is at most j and of is at most k. While the problem of verifying whether a graph is (0, 0)-colorable is easy, the similar problem with in place of (0, 0) is NP-complete for all nonnegative j and k with . Let denote the supremum of all x such that for some constant every graph G with girth g and for every is -colorable. It was proved recently that . In a companion paper, we find the exact value . In this article, we show that increasing g from 5 further on does not increase much. Our constructions show that for every g, . We also find exact values of for all g and all k >= 2 j + 2.(C) 2015 Wiley Periodicals, Inc. J. Graph Theory 81: 403-413, 2016