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

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) -> 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), . . . , x(n), x(0)), in an H-colored graph, is an H-cycle if and only if (c(x(0)x(1)), c(x(1)x(2)), . . . , c(x(n)x(0)), c(x(0)x(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 N.