On the existence of cycles with restrictions in the color transitions in edge-colored complete graphs (Q6564830)

Given a graph \(G\) and another graph \(H\), the \(H\)-coloring of \(G\) is any function \(c: E(G)\rightarrow V(H)\). \(G\) is \(H\)-colored for any such fixed \(c\). A cycle \(x_0,x_1,\dots,x_k,x_0\) is an \(H\)-cycle if and only if \(c(x_0x_1),c(x_1x_2),\dots,c(x_kx_0),c(x_0x_1)\) is a walk in \(H\). In this paper, the authors prove that every \(v\in V(G)\) belongs to an \(H\)-cycle of length at least \(5\), where \(G\) is an \(H\)-colored complete graph with \(n=|V(G)|\geq 13\) satisfying the two following local conditions:\N\begin{itemize}\N\item for every \(x\in V(G)\), \(G_x\) is a complete \(k_x\)-partite graph for some natural number \(k_x\geq\frac{n+1}{2}\), where \(V(G_x)=\{e\in E(G):x\in e\}\) and \(E(G_x)=\{ef: e\in V(G_x), f\in V(G_x), e\neq f, c(e)c(f)\in E(H) \}\),\N\item there is no cycle \(xyzx\) in \(G\) such that \(c(xy)c(yz)\notin E(H)\) and \(c(yz)c(zx)\notin E(H)\).\N\end{itemize}\NAlso, an infinite family of graphs satisfying the assumptions of the theorem is presented.\N\NIt is worth noticing that a simple consequence of the above is a previous result by \textit{S. Fujita} and \textit{C. Magnant} [Discrete Appl. Math. 159, No. 14, 1391--1397 (2011; Zbl 1228.05150)] stating that if in a complete edge-colored graph \(G\) with \(n\geq 13\) the color degree \(\delta^c(v)\) of every vertex \(v\in V(G)\) (i.e., the number of colors used on the incident edges) satisfies \(\delta^c(v)\geq\frac{n+1}{2}\), then each \(v\in V(G)\) belongs to a properly colored cycle of length at least \(5\). For that reason, the authors suppose that the second local condition in the theorem can be actually omitted.