On sum edge-coloring of regular, bipartite and split graphs

TL;DR

This paper studies the computational complexity of sum edge-coloring in regular, bipartite, and split graphs, providing approximation algorithms, NP-completeness results, and bounds for specific graph classes.

Contribution

It introduces a polynomial-time approximation algorithm for r-regular graphs and establishes NP-completeness for bipartite graphs with maximum degree 3, along with bounds for split graphs.

Findings

Approximation algorithm with ratio (1+2r/(r+1)^2) for r-regular graphs

NP-completeness of sum edge-coloring for bipartite graphs with max degree 3

Upper bounds for edge-chromatic sum in certain split graphs

Abstract

An edge-coloring of a graph $G$ with natural numbers is called a sum edge-coloring if the colors of edges incident to any vertex of $G$ are distinct and the sum of the colors of the edges of $G$ is minimum. The edge-chromatic sum of a graph $G$ is the sum of the colors of edges in a sum edge-coloring of $G$. It is known that the problem of finding the edge-chromatic sum of an $r$-regular ($r\geq 3$) graph is $NP$-complete. In this paper we give a polynomial time $(1+\frac{2r}{(r+1)^{2}})$-approximation algorithm for the edge-chromatic sum problem on $r$-regular graphs for $r\geq 3$. Also, it is known that the problem of finding the edge-chromatic sum of bipartite graphs with maximum degree 3 is $NP$-complete. We show that the problem remains $NP$-complete even for some restricted class of bipartite graphs with maximum degree 3. Finally, we give upper bounds for the edge-chromatic sum of some split graphs.