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

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(vₖ₋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.