The Edge-Distinguishing Chromatic Number of Petal Graphs, Chorded Cycles, and Spider Graphs

TL;DR

This paper determines the minimum vertex coloring number needed to uniquely identify edges in specific complex graphs like petal graphs, chorded cycles, and four-legged spider graphs, expanding previous knowledge.

Contribution

It introduces new methods to compute the edge-distinguishing chromatic number for complex graphs using graph embedding techniques.

Findings

EDCN of petal graphs with two petals and a loop is established.

EDCN of cycles with one chord is determined.

EDCN of spider graphs with four legs is calculated.

Abstract

The edge-distinguishing chromatic number (EDCN) of a graph $G$ is the minimum positive integer $k$ such that there exists a vertex coloring $c:V(G)\to\{1,2,\dotsc,k\}$ whose induced edge labels $\{c(u),c(v)\}$ are distinct for all edges $uv$. Previous work has determined the EDCN of paths, cycles, and spider graphs with three legs. In this paper, we determine the EDCN of petal graphs with two petals and a loop, cycles with one chord, and spider graphs with four legs. These are achieved by graph embedding into looped complete graphs.