Color-blind index in graphs of very low degree

TL;DR

This paper introduces the concept of the color-blind index in graphs, explores its computational complexity, and establishes connections to hypergraph colorability, revealing new insights into graph coloring problems.

Contribution

It defines the color-blind index, proves NP-completeness for certain cases, and links it to hypergraph 2-colorability, advancing understanding of graph coloring complexities.

Findings

Determining if the color-blind index is at most 2 is NP-complete.

The color-blind index relates to 2-colorable regular hypergraphs.

Characterizes finiteness of the index for certain 3-regular graphs.

Abstract

Let $c:E(G)\to [k]$ be an edge-coloring of a graph $G$, not necessarily proper. For each vertex $v$, let $\bar{c}(v)=(a_1,\ldots,a_k)$, where $a_i$ is the number of edges incident to $v$ with color $i$. Reorder $\bar{c}(v)$ for every $v$ in $G$ in nonincreasing order to obtain $c^*(v)$, the color-blind partition of $v$. When $c^*$ induces a proper vertex coloring, that is, $c^*(u)\neq c^*(v)$ for every edge $uv$ in $G$, we say that $c$ is color-blind distinguishing. The minimum $k$ for which there exists a color-blind distinguishing edge coloring $c:E(G)\to [k]$ is the color-blind index of $G$, denoted $\operatorname{dal}(G)$. We demonstrate that determining the color-blind index is more subtle than previously thought. In particular, determining if $\operatorname{dal}(G) \leq 2$ is NP-complete. We also connect the color-blind index of a regular bipartite graph to 2-colorable regular hypergraphs and characterize when $\operatorname{dal}(G)$ is finite for a class of 3-regular graphs.