A locally irregular decomposition of a graph \(G\) is a collection \(\{ G_1, \ldots ,G_k \}\) of edge-disjoint subgraphs of \(G\) such that \(E(G_1) \cup \cdots \cup E(G_k) = E(G)\) and every \(G_i\) is locally irregular, i.e., \(d_{G_i}(u) \ne d_{G_i}(v)\) for every edge \(uv \in E(G_i)\) . The smallest \(k\) for which \(G\) has a locally irregular decomposition of size \(k\) is called the irregular chromatic index of \(G\) , and is denoted by \(\chi '_{\textrm{irr}}(G)\) . A blow-up of a graph \(H\) with \(V(H) = \{ v_1, \ldots ,v_h \}\) is a graph \(G\) obtained by replacing each vertex \(v_i\) of \(H\) with an independent set \(V_i\) , and each edge \(v_iv_j\) with all edges between \(V_i\) and \(V_j\) . We prove that, for any cycle \(C_k\) of length \(k \ge 3\) , \(\chi '_{\textrm{irr}}(G) \le 2\) for any blow-up \(G\) of \(C_k\) with at least one class of size greater than 1.
