Graph Classes with Locally Irregular Chromatic Index at most 4

Abstract

A graph G is said to be locally irregular if each pair of adjacent vertices have different degrees in G. A collection of edge disjoint subgraphs $(G_1,\ldots ,G_k)$ of G is called a k-locally irregular decomposition of G if $(E(G_1),\ldots ,E(G_k))$ is an edge partition of G and each $G_i$ is locally irregular for $i\in \{1,\ldots ,k\}$. The locally irregular chromatic index of G, denoted by $\chi '_{irr}(G)$, is the smallest integer k such that G can be decomposed into k locally irregular subgraphs. A graph G is said to be decomposable if $\chi '_{irr}(G)$ is finite, otherwise, G is exceptional. The Local Irregularity Conjecture states that all connected graphs admit a 3-locally irregular decomposition except for odd paths, odd cycles, and a certain subclass of cacti. Recently, Sedlar and Škrekovski showed that there exists a graph G which is a cactus such that $\chi '_{irr}(G)=4$. In this paper, we mainly prove that if G is a decomposable cactus, then $\chi '_{irr}(G)\le 4$; if G is a decomposable cactus without nontrivial cut edges, then $\chi '_{irr}(G)\le 3$. In addition, we show that in a decomposable subcubic graph G if each vertex of degree 3 lies on a triangle, then $\chi '_{irr}(G)\le 3$. By establishing algorithms, we obtain $\chi '_{irr}(K_n-C_\ell )\le 3$ for $3\le \ell \le n-1$.