A Note On Vertex Distinguishing Edge colorings of Trees

TL;DR

This paper investigates the vertex distinguishing edge coloring of trees, establishing exact values of the vdec chromatic number based on degree distributions and diameter, and relates it to equitable colorings under certain conditions.

Contribution

It provides exact formulas for the vdec chromatic number of trees depending on degree counts and diameter, and links it to equitable vdec colorings.

Findings

For trees with $n_2(T) \,\leq\, n_1(T)$ and diameter 3 or specific diameter 4 trees, $\,\chi\'_s(T) = n_1(T)+1$.

Otherwise, $\,\chi\'_s(T) = n_1(T)$.

When $|E(T)| \leq 2(n_1(T)+1)$, the equitable vdec chromatic number equals the vdec chromatic number.

Abstract

A proper edge coloring of a simple graph $G$ is called a vertex distinguishing edge coloring (vdec) if for any two distinct vertices $u$ and $v$ of $G$, the set of the colors assigned to the edges incident to $u$ differs from the set of the colors assigned to the edges incident to $v$. The minimum number of colors required for all vdecs of $G$ is denoted by $\chi\,'_s(G)$ called the vdec chromatic number of $G$. Let $n_d(G)$ denote the number of vertices of degree $d$ in $G$. In this note, we show that a tree $T$ with $n_2(T)\leq n_1(T)$ holds $\chi\,'_s(T)=n_1(T)+1$ if its diameter $D(T)=3$ or one of two particular trees with $D(T) =4$, and $\chi\,'_s(T)=n_1(T)$ otherwise; furthermore $\chi\,'_{es}(T)=\chi\,'_s(T)$ when $|E(T)|\leq 2(n_1(T)+1)$, where $\chi\,'_{es}(T)$ is the equitable vdec chromatic number of $T$.