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arXiv · 1606.06333

Tight Bounds on the Complexity of Semi-Equitable Coloring of Cubic and Subcubic Graphs

H. Furmańczyk·M. Kubale
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Abstract

A $k$-coloring of a graph $G=(V,E)$ is called semi-equitable if there exists a partition of its vertex set into independent subsets $V_1,\ldots,V_k$ in such a way that $|V_1| \notin \{\lceil |V|/k\rceil, \lfloor |V|/k \rfloor\}$ and $||V_i|-|V_j|| \leq 1$ for each $i,j=2,\ldots,k$. The color class $V_1$ is called non-equitable. In this note we consider the complexity of semi-equitable $k$-coloring, $k\geq 4$, of the vertices of a cubic or subcubic graph $G$. In particular, we show that, given a $n$-vertex subcubic graph $G$ and constants $\epsilon > 0$, $k \geq 4$, it is NP-complete to obtain a semi-equitable $k$-coloring of $G$ whose non-equitable color class is of size $s$ if $s \geq n/3+\epsilon n$, and it is polynomially solvable if $s \leq n/3$.

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H. Furmańczyk, M. Kubale. 2016-06-20. Tight Bounds on the Complexity of Semi-Equitable Coloring of Cubic and Subcubic Graphs. https://arxiv.org/abs/1606.06333

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