Consider the following edge-coloring of a graph G. Let H be a graph possibly with loops, an H-coloring of a graph G is defined as a function \(c : E(G) \rightarrow V(H).\) We will say that G is an H-colored graph whenever we are taking a fixed H-coloring of G. A cycle \((x_0,x_1,\ldots ,x_n,x_0),\) in an H-colored graph, is an H-cycle if and only if \((c(x_0x_1),c(x_1x_2),\ldots , c(x_nx_0),\) \(c(x_0x_1))\) is a walk in H. Notice that the graph H determines what color transitions are allowed in a cycle in order to be an H-cycle, in particular, when H is a complete graph without loops, every H-cycle is a properly colored cycle. In this paper, we give conditions on an H-colored complete graph G, with local restrictions, implying that every vertex of G is contained in an H-cycle of length at least 5. As a consequence, we obtain a previous result about properly colored cycles. Finally, we show an infinite family of H-colored complete graphs fulfilling the conditions of the main theorem, where the graph H is not a complete k-partite graph for any k in \({\mathbb {N}}.\)
