Complexity of tree-coloring interval graphs equitably

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 $k\geq \lceil(\Delta(G)+1)/2\rceil$, 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 $K_{1,r}$-free interval graphs with $r\geq 4$, 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 Equitable List Arboricity of Graphs

Equitable list arboricity, introduced by Zhang in 2016, generalizes the notion of equitable list coloring by requiring the subgraph induced by each color class to be acyclic (instead of edgeless) in addition to the usual upper bound on the size of each color class. Graph $G$ is equitably $k$-list arborable if an equitable, arborable list coloring of $G$ exists for every list assignment for $G$ that associates with each vertex in $G$ a list of $k$ available colors. Zhang conjectured that any graph $G$ is equitably $k$-list arborable for each $k$ satisfying $k \geq \lceil (1+\Delta(G))/2 \rceil$. We verify this conjecture for powers of cycles by applying a new lemma which is a general tool for extending partial equitable, arborable list colorings. We also propose a stronger version of Zhang's Conjecture for certain connected graphs: any connected graph $G$ is equitably $k$-list arborable for each $k$ satisfying $k \geq \lceil \Delta(G)/2 \rceil$ provided $G$ is neither a cycle nor a complete graph of odd order. We verify this stronger version of Zhang's Conjecture for powers of paths, 2-degenerate graphs, and certain other graphs. We also show that if $G$ is equitably $k$-list arborable it does not necessarily follow that $G$ is equitably $(k+1)$-list arborable which addresses a question of Drgas-Burchardt, Furmanczyk, and Sidorowicz (2018).Comment: 20 page