Cycles of length 3 and 4 in edge-colored complete graphs with restrictions in the color transitions

Abstract Let G be a graph and H a graph possibly with loops. We will say that a graph G is an H -colored graph if and only if there exists a function c: E (G) V (H) c: E (G) ⟶ V (H). A cycle (v₁, , vₖ, v₁) (v 1, …, v k, v 1) is an H -cycle if and only if (c (v₁ v₂), , c (v₊-₁vₖ), (c (v 1 v 2), …, c (v k - 1 v k), c (vₖv₁), c (v₁ v₂) ) c (v k v 1), c (v 1 v 2) ) is a walk in H. Whenever H is a complete graph without loops, an H -cycle is a properly colored cycle. In this paper, we work with an H -colored complete graph, namely G, with local restrictions given by an auxiliary graph, and we show sufficient conditions implying that every vertex in V (G) is contained in an H -cycle of length 3 (respectively 4). As a consequence, we obtain some well-known results in the theory of properly colored walks.