On Local Irregularity Conjecture for 2-multigraphs

A multigraph in which adjacent vertices have different degrees is called locally irregular. The locally irregular edge coloring is an edge coloring of a multigraph \(G\) in which every color induces a locally irregular submultigraph of \(G\). We denote by \(\operatorname{lir}(G)\) the locally irregular chromatic index of a multigraph \(G\), which is the smallest number of colors required in a locally irregular edge coloring of \(G\), given that such a coloring of \(G\) exists. By \(^2G\) we denote a 2-multigraph obtained from a simple graph \(G\) by doubling each its edge. In 2022 Grzelec and Woźniak conjectured that \(\operatorname{lir}(^2G) \leq 2\) for every connected simple graph \(G\) different from \(K_2\); the conjecture is known as Local Irregularity Conjecture for 2-multigraphs. In this paper, we prove this conjecture in the case of regular graphs, split graphs, and some particular families of subcubic graphs. Moreover, we provide a constant upper bound on the locally irregular chromatic index of planar 2-multigraphs (except for \(^2K_2\)), and we obtain a better constant upper bound on \(\operatorname{lir}(^2G)\) if \(G\) is a simple subcubic graph different from \(K_2\). In the proofs, special decompositions of graphs and the relation of Local Irregularity Conjecture to the well-known 1-2-3 Conjecture are utilized.