Graph Classes with Locally Irregular Chromatic Index at most 4

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 $$(G_1,\ldots ,G_k)$$ ( G 1 , … , G k ) of G is called a k-locally irregular decomposition of G if $$(E(G_1),\ldots ,E(G_k))$$ ( E ( G 1 ) , … , E ( G k ) ) is an edge partition of G and each $$G_i$$ G i is locally irregular for $$i\in \{1,\ldots ,k\}$$ i ∈ { 1 , … , k } . The locally irregular chromatic index of G, denoted by $$\chi '_{irr}(G)$$ χ 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 $$\chi '_{irr}(G)$$ χ 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 $$\chi '_{irr}(G)=4$$ χ irr ′ ( G ) = 4 . In this paper, we mainly prove that if G is a decomposable cactus, then $$\chi '_{irr}(G)\le 4$$ χ irr ′ ( G ) ≤ 4 ; if G is a decomposable cactus without nontrivial cut edges, then $$\chi '_{irr}(G)\le 3$$ χ irr ′ ( G ) ≤ 3 . In addition, we show that in a decomposable subcubic graph G if each vertex of degree 3 lies on a triangle, then $$\chi '_{irr}(G)\le 3$$ χ irr ′ ( G ) ≤ 3 . By establishing algorithms, we obtain $$\chi '_{irr}(K_n-C_\ell )\le 3$$ χ irr ′ ( K n - C ℓ ) ≤ 3 for $$3\le \ell \le n-1$$ 3 ≤ ℓ ≤ n - 1 .