Vertex-distinguishing and sum-distinguishing edge coloring of regular graphs

TL;DR

This paper investigates vertex-distinguishing and sum-distinguishing edge colorings of regular graphs, establishing new bounds for the minimum number of colors needed in these colorings for large, dense regular graphs.

Contribution

The paper improves bounds on the chromatic indices for regular graphs, extending previous results and introducing new edge coloring techniques of independent interest.

Findings

+2 colors suffice when degree is close to half the number of vertices

+2 colors suffice for sum-distinguishing when degree is at least two-thirds of vertices

Introduces novel edge coloring methods applicable beyond the studied cases

Abstract

Given an integer $k\ge1$, an edge-$k$-coloring of a graph $G$ is an assignment of $k$ colors $1,\ldots,k$ to the edges of $G$ such that no two adjacent edges receive the same color. A vertex-distinguishing (resp. sum-distinguishing) edge-$k$-coloring of $G$ is an edge-$k$-coloring such that for any two distinct vertices $u$ and $v$, the set (resp. sum) of colors taken from all the edges incident with $u$ is different from that taken from all the edges incident with $v$. The vertex-distinguishing chromatic index (resp. sum-distinguishing chromatic index), denoted $\chi'_{vd}(G)$ (resp. $\chi'_{sd}(G)$), is the smallest value $k$ such that $G$ has a vertex-distinguishing-edge-$k$-coloring (resp. sum-distinguishing-edge-$k$-coloring). Let $G$ be a $d$-regular graph on $n$ vertices, where $n$ is even and sufficiently large. We show that $\chi'_{vd}(G) =d+2$ if $d$ is arbitrarily close to $n/2$ from above, and $\chi'_{sd}(G) =d+2$ if $d\ge \frac{2n}{3}$. Our first result strengthens a result of Balister et al. in 2004 for such class of regular graphs, and our second result constitutes a significant advancement in the field of sum-distinguishing edge coloring. To achieve these results, we introduce novel edge coloring results which may be of independent interest.