Hardness and algorithms of equitable tree-coloring problem in chordal graphs

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 differs by at most one. In this work, we show that every interval graph G has an equitable tree-k-coloring for any integer k≥⌈(Δ(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. On the positive side, we propose a quadratic 2-approximation algorithm for the equitable tree-coloring problem in chordal graphs. Moreover, it is proved that there is no α-approximation algorithm for any α<32.