Color neighborhood union conditions for proper edge-pancyclicity of edge-colored complete graphs
An edge-colored graph \(G^c\) consists of a graph \(G\) and a coloring mapping \(c\) which assigns colors to edges of \(G\). For a vertex \(v \in V(G^c)\), the color neighborhood of \(v\), denoted \(CN(v)\), is the set of colors used to color the neighbors of \(v\) in \(G\). The research investigates the sufficient conditions in terms of color neighborhood union to assure the existences of proper colored cycles with various lengths. The main results prove that for an edge-colored complete graph \(K_n^c\) on \(n\) vertices, for any integer \(k \ge 3\), if \(n\) is sufficiently large and if for any pair of distinct vertices \(u\) and \(v\), \(|CN(u) \cup CN(v)| \ge n\), then \(K_n^c\) has a properly colored cycle of length \(k\). Furthermore, if, in addition, \(K_n^c\) does not have a monochromatic triangle, then for any vertex \(v \in V(G^c)\) and for any integer \(k\) with \(3 \le k \le n\), there exists a properly colored cycle \(C\) of length \(k\) containing \(v\); and for any distinct vertices \(u,v\) and for any integer \(\ell\) with \(1 \le \ell \le n-1\), \(G^c\) contains a properly colored \((u,v)\)-path of length \(\ell\). A few open problems are raised in the concluding remark section of the paper, which seem quite interesting.
scientific article; zbMATH DE number 7439558
scientific article