A study on edge coloring and edge sum coloring of integral sum graphs

TL;DR

This paper explores edge coloring and edge sum coloring in integral sum graphs, establishing properties of edge-sum classes, and comparing edge chromatic numbers across different graph families with new labeling methods.

Contribution

It introduces the concept of edge sum chromatic number in integral sum graphs and compares it with the edge chromatic number for various graph classes, providing new insights and labelings.

Findings

Edge-sum classes partition the edge set of integral sum graphs.

Edge chromatic number equals edge sum chromatic number for certain graphs like $G_{0,s}$ and $S_n$.

For other graphs like $G_{r,s}$, these numbers differ.

Abstract

Frank Harary introduced the concept of integral sum graph. A graph $G$ is an \emph{ integral sum graph} if its vertices can be labeled with distinct integers so that $e = uv$ is an edge of $G$ if and only if the sum of the labels on vertices $u$ and $v$ is also a label in $G.$ For any non-empty set of integers $S$, let $G^+(S)$ denote the integral sum graph on the set $S$. In $G^+(S)$, we define an \emph{edge-sum class} as the set of all edges each with same edge sum number and call $G^+(S)$ an \emph{edge sum color graph} if each edge-sum class is considered as an edge color class of $G^+(S)$. The number of distinct edge-sum classes of $G^+(S)$ is called its \emph{ edge sum chromatic number}. The main results of this paper are (i) the set of all edge-sum classes of an integral sum graph partitions its edge set; (ii) the edge chromatic number and the edge sum chromatic number are equal for the integral sum graphs $G_{0,s}$ and $S_n$, Star graph of order $n$, whereas it is not in the case of $G_{r,s} = G^+([r,s])$, $r < 0 < s$, $n,s \geq 2$, $n,r,s\in\mathbb{N}$. We also obtain an interesting integral sum labeling of Star graphs.