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

TL;DR

This paper investigates the computational complexity of semi-equitable coloring in cubic and subcubic graphs, establishing NP-completeness for certain class sizes and polynomial solvability for others, thus providing tight bounds on the problem.

Contribution

It proves NP-completeness for semi-equitable k-coloring with large non-equitable class sizes and polynomial solvability for smaller sizes in cubic and subcubic graphs.

Findings

NP-complete for non-equitable class size s ≥ n/3 + εn

Polynomial time solvable for s ≤ n/3

Provides tight bounds on semi-equitable coloring complexity

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 $s \leq n/3$.