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.

中文翻译：

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树着色区间图的复杂性公平

一个图的公平树-$k$-着色是一个顶点$k$-着色，使得每个颜色类都产生一个森林，并且任何两个颜色类的大小最多相差一个。在这项工作中，我们证明了每个区间图 $G$ 对任何整数 $k\geq \lceil(\Delta(G)+1)/2\rceil$ 都有一个公平的树-$k$-着色，解决了一个猜想Wu、Zhang 和 Li (2013) 的区间图，此外，给出了一个线性时间算法，用于确定适当的区间图是否承认给定整数 $k$ 的公平树 $k$-着色。对于分裂图的不相交并集，或 $K_{1,r}$-free 区间图与 $r\geq 4$，我们证明 $W[1]$-很难决定是否存在公平树- $k$-coloring 当分别按颜色数或树宽、颜色数和最大度数参数化时。