On decomposing multigraphs into locally irregular submultigraphs

Abstract

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 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 K2, the multigraph 2G obtained from G by doubling each edge admits lir(2G)≤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.