On decomposing multigraphs into locally irregular submultigraphs

A locally irregular multigraph is a multigraph whose adjacent vertices have distinct degrees. The locally irregular edge coloring is an edge coloring of a multigraph $G$ such that every color induces a locally irregular submultigraph of $G$. We say that a multigraph $G$ is locally irregular colorable if it admits a locally irregular edge coloring and we denote by ${\rm lir}(G)$ the locally irregular chromatic index of $G$, which is the smallest number of colors required in a locally irregular edge coloring of a locally irregular colorable multigraph $G$. We conjecture that for every connected graph $G$, which is not isomorphic to $K_2$, multigraph $^2G$ obtained from $G$ by doubling each edge admits ${\rm lir}(^2G)\leq 2$. This concept is closely related to the well known 1-2-3 Conjecture, Local Irregularity Conjecture, (2, 2) Conjecture and other similar problems concerning edge colorings. We show this conjecture holds for graph classes like paths, cycles, wheels, complete graphs, complete $k$-partite graphs and bipartite graphs. We also prove the general bound for locally irregular chromatic index for all 2-multigraphs using our result for bipartite graphs.

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