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

Abstract 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.