The strict neighbor-distinguishing index of simple graphs with maximum degree five

<p> Suppose that \(\phi\) is a proper edge-\(k\)-coloring of the graph \(G\). For a vertex \(v \in V(G)\), let \(C_\phi(v)\) denote the set of colors assigned to the edges incident with \(v\). The proper edge-\(k\)-coloring \(\phi\) of \(G\) is strict neighbor-distinguishing if for any adjacent vertices \(u\) and \(v\), \(C_\phi(u) \varsubsetneq C_\phi(v)\) and \(C_\phi(v) \varsubsetneq C_\phi(u)\). The strict neighbor-distinguishing index, denoted \(\chi'_{snd}(G)\), is the minimum integer \(k\) such that \(G\) has a strict neighbor-distinguishing edge-\(k\)-coloring. In this paper we prove that if \(G\) is a simple graph with maximum degree five, then \(\chi'_{snd}(G) \leq 12\). </p>