  We say that a graph or a multigraph is locally irregular if adjacent vertices
have different degrees. The locally irregular edge coloring is an edge coloring
of a (multi)graph $G$ such that every color induces a locally irregular
sub(multi)graph of $G$. We denote by ${\rm lir}(G)$ the locally irregular
chromatic index of $G$, which is the smallest number of colors required in a
locally irregular edge coloring of $G$ if such a coloring exists. We state the
following new problem. Let $G$ be a connected graph that is not isomorphic to
$K_2$ or $K_3$. What is the minimum number of edges of $G$ whose doubling
yields a multigraph which is locally irregular edge colorable using at most two
colors with no multiedges colored with two colors? This problem is closely
related to the Local Irregularity Conjecture for graphs, (2, 2)-Conjecture,
Local Irregularity Conjecture for 2-multigraphs and other similar concepts
concerning edge colorings. We solve this problem for graph classes like paths,
cycles, trees, complete graphs, complete $k$-partite graphs, split graphs and
powers of cycles. Our solution of this problem for complete $k$-partite graphs
($k>1$) and powers of cycles which are not complete graphs shows that the
locally irregular chromatic index is equal to two for these graph classes. We
also consider this problem for some special families of cacti and prove that
the minimum number of edges in a graph whose doubling yields a multigraph which
has such coloring does not have a constant upper bound not only for locally
irregular uncolorable cacti.
