  A graph is locally irregular if the degrees of the end-vertices of every edge
are distinct. An edge coloring of a graph G is locally irregular if every color
induces a locally irregular subgraph of G. A colorable graph G is any graph
which admits a locally irregular edge coloring. The locally irregular chromatic
index X'irr(G) of a colorable graph G is the smallest number of colors required
by a locally irregular edge coloring of G. The Local Irregularity Conjecture
claims that all colorable graphs require at most 3 colors for locally irregular
edge coloring. Recently, it has been observed that the conjecture does not hold
for the bow-tie graph B [7]. Cacti are important class of graphs for this
conjecture since B and all non-colorable graphs are cacti. In this paper we
show that for every colorable cactus graph G != B it holds that X'irr(G) <= 3.
This makes us to believe that B is the only colorable graph with X'irr(B) > 3,
and consequently that B is the only counterexample to the Local Irregularity
Conjecture.
