Exact defective colorings of graphs

TL;DR

This paper introduces the concept of exact $(k,d)$-colorings in graphs, explores their properties for various graph classes, and proves NP-completeness of the coloring decision problem for all $d \\geq 1$ and $k \\geq 2$.

Contribution

It defines and analyzes the exact $d$-defective chromatic number, providing bounds, exact values for specific graphs, and polynomial algorithms for certain graph classes, along with complexity results.

Findings

Exact values for cycles, trees, and complete graphs.

Polynomial algorithms for cactus and block graphs.

NP-completeness for $d$-EXACT DEFECTIVE $k$-COLORING when $d \\geq 1$ and $k \\geq 2$.

Abstract

An exact $(k,d)$-coloring of a graph $G$ is a coloring of its vertices with $k$ colors such that each vertex $v$ is adjacent to exactly $d$ vertices having the same color as $v$. The exact $d$-defective chromatic number, denoted $\chi_d^=(G)$, is the minimum $k$ such that there exists an exact $(k,d)$-coloring of $G$. In an exact $(k,d)$-coloring, which for $d=0$ corresponds to a proper coloring, each color class induces a $d$-regular subgraph. We give basic properties for the parameter and determine its exact value for cycles, trees, and complete graphs. In addition, we establish bounds on $\chi_d^=(G)$ for all relevant values of $d$ when $G$ is planar, chordal, or has bounded treewidth. We also give polynomial-time algorithms for finding certain types of exact $(k,d)$-colorings in cactus graphs and block graphs. Our main result is on the computational complexity of $d$-EXACT DEFECTIVE $k$-COLORING in which we are given a graph $G$ and asked to decide whether $\chi_d^=(G) \leq k$. Specifically, we prove that the problem is NP-complete for all $d \geq 1$ and $k \geq 2$.