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A graph is locally irregular if the degrees of the end-vertices of every edgeare distinct. An edge coloring of a graph G is locally irregular if every colorinduces a locally irregular subgraph of G. A colorable graph G is any graphwhich admits a locally irregular edge coloring. The locally irregular chromaticindex X'irr(G) of a colorable graph G is the smallest number of colors requiredby a locally irregular edge coloring of G. The Local Irregularity Conjectureclaims that all colorable graphs require at most 3 colors for a locallyirregular edge coloring. Recently, it has been observed that the conjecturedoes not hold for the bow-tie graph B, since B is colorable and requires atleast 4 colors for a locally irregular edge coloring. Since B is a cactus graphand all non-colorable graphs are also cacti, this seems to be a relevant classof graphs for the Local Irregularity Conjecture. In this paper we establishthat X'irr(G)<= 4 for all colorable cactus graphs.

A note on the locally irregular edge colorings of cacti