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A multigraph is locally irregular if the degrees of the end-vertices of everymultiedge are distinct. The locally irregular coloring is an edge coloring of amultigraph $G$ such that every color induces a locally irregular submultigraphof $G$. A locally irregular colorable multigraph $G$ is any multigraph whichadmits a locally irregular coloring. We denote by ${\rm lir}(G)$ the locallyirregular chromatic index of a multigraph $G$, which is the smallest number ofcolors required in the locally irregular coloring of the locally irregularcolorable multigraph $G$. In case of graphs the definitions are similar. TheLocal Irregularity Conjecture for 2-multigraphs claims that for every connectedgraph $G$, which is not isomorphic to $K_2$, multigraph $^2G$ obtained from $G$by doubling each edge satisfies ${\rm lir}(^2G)\leq 2$. We show this conjecturefor cacti. This class of graphs is important for the Local IrregularityConjecture for 2-multigraphs and the Local Irregularity Conjecture which claimsthat every locally irregular colorable graph $G$ satisfies ${\rm lir}(G)\leq3$. At the beginning it has been observed that all not locally irregularcolorable graphs are cacti. Recently it has been proved that there is only onecactus which requires 4 colors for a locally irregular coloring and thereforethe Local Irregularity Conjecture was disproved.

Local Irregularity Conjecture for 2-multigraphs versus cacti