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). 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 (x₀,x₁,… ,xₙ,x₀), ( x 0 , x 1 , … , x n , x 0 ) , in an H -colored graph, is an H - cycle if and only if (c(x₀x₁),c(x₁x₂),… , c(xₙx₀), ( c ( x 0 x 1 ) , c ( x 1 x 2 ) , … , c ( x n x 0 ) , c(x₀x₁)) 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. N .