Local irregularity conjecture for 2-multigraphs versus cacti

Abstract

A multigraph is locally irregular if the degrees of the end-vertices of every multiedge are distinct. The locally irregular coloring is an edge coloring of a multigraph \(G\) such that every color induces a locally irregular submultigraph of \(G\). A locally irregular colorable multigraph \(G\) is any multigraph which admits a locally irregular coloring. We denote by \(\textrm{lir}(G)\) the locally irregular chromatic index of a multigraph \(G\), which is the smallest number of colors required in the locally irregular coloring of the locally irregular colorable multigraph \(G\). In case of graphs the definitions are similar. The Local Irregularity Conjecture for 2-multigraphs claims that for every connected graph \(G\), which is not isomorphic to \(K_2\), multigraph \(^2G\) obtained from \(G\) by doubling each edge satisfies \(\textrm{lir}(^2G)\leq 2\). We show this conjecture for cacti. This class of graphs is important for the Local Irregularity Conjecture for 2-multigraphs and the Local Irregularity Conjecture which claims that every locally irregular colorable graph \(G\) satisfies \(\textrm{lir}(G)\leq 3\). At the beginning it has been observed that all not locally irregular colorable graphs are cacti. Recently it has been proved that there is only one cactus which requires 4 colors for a locally irregular coloring and therefore the Local Irregularity Conjecture was disproved.