  A locally irregular multigraph is a multigraph whose adjacent vertices have
distinct degrees. The locally irregular edge coloring is an edge coloring of a
multigraph $G$ such that every color induces a locally irregular submultigraph
of $G$. We say that a multigraph $G$ is locally irregular colorable if it
admits a locally irregular edge coloring and 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 a locally irregular
colorable multigraph $G$. We conjecture that for every connected graph $G$,
which is not isomorphic to $K_2$, multigraph $^2G$ obtained from $G$ by
doubling each edge admits ${\rm lir}(^2G)\leq 2$. This concept is closely
related to the well known 1-2-3 Conjecture, Local Irregularity Conjecture, (2,
2) Conjecture and other similar problems concerning edge colorings. We show
this conjecture holds for graph classes like paths, cycles, wheels, complete
graphs, complete $k$-partite graphs and bipartite graphs. We also prove the
general bound for locally irregular chromatic index for all 2-multigraphs using
our result for bipartite graphs.
