Remarks on the Local Irregularity Conjecture

A locally irregular graph is a in which the end vertices of every edge have distinct degrees. coloring G any such that each colors induces subgraph G. colorable if it allows coloring. The chromatic index G, denoted by χirr′(G), smallest number used local irregularity conjecture claims all graphs, except odd-length paths, cycles and certain class cacti are three colors. As valid for graphs with large minimum degree non-colorable vertex disjoint cacti, we study rather sparse graphs. In this paper, give cactus B contradicts conjecture, i.e., χirr′(B)=4. Nevertheless, show holds unicyclic cycles.