On a new problem about the local irregularity of graphs

Abstract

A graph/multigraph $G$ is locally irregular if endvertices of every its edge possess different degrees. The locally irregular edge coloring of $G$ is its edge coloring with the property that every color induces a locally irregular sub(multi)graph of $G$; if such a coloring of $G$ exists, the minimum number of colors to color $G$ in this way is the locally irregular chromatic index of $G$ (denoted by ${\rm lir}(G)$). We state the following new problem: given a connected graph $G$ distinct from $K_2$ or $K_3$, what is the minimum number of edges of $G$ to be doubled such that the resulting multigraph is locally irregular edge colorable (with no monochromatic multiedges) using at most two colors? This problem is closely related to several open conjectures (like the Local Irregularity Conjecture for graphs and 2-multigraphs, or (2, 2)-Conjecture) and other similar edge coloring concepts. We present the solution of this problem for several graph classes: paths, cycles, trees, complete graphs, complete $k$-partite graphs, split graphs and powers of cycles. Our solution for complete $k$-partite graphs ($k>1$) and powers of cycles (which are not complete graphs) shows that, in this case, the locally irregular chromatic index equals 2. We also consider this problem for special families of cacti and prove that the minimum number of edges in a graph whose doubling yields an local irregularly colorable multigraph does not have a constant upper bound not only for locally irregular uncolorable cacti.