Complexity of tree-coloring interval graphs equitably

Abstract

An equitable tree-k-coloring of a graph is a vertex k-coloring such that each color class induces a forest and the size of any two color classes differ by at most one. In this work, we show that every interval graph G has an equitable tree-k-coloring for any integer k ≥ ⌈(∆(G) + 1)/2⌉, solving a conjecture of Wu, Zhang and Li (2013) for interval graphs, and furthermore, give a linear-time algorithm for determining whether a proper interval graph admits an equitable tree-k-coloring for a given integer k. For disjoint union of split graphs, or K1,r-free interval graphs with r ≥ 4, we prove that it is W [1]hard to decide whether there is an equitable tree-k-coloring when parameterized by number of colors, or by treewidth, number of colors and maximum degree, respectively.