Vertex-pancyclism in edge-colored complete graphs with restrictions in color transitions

Let H be a graph possibly with loops.Let G be a simple graph.We say that G is an H-colored graph whenever every edge of G has assigned a vertex of H as a color.A cycle C in an H-colored graph G is an H-cycle if and only if the colors of consecutive edges in C are adjacent vertices in H, including the last and first edges of C.An H-colored graph G is said to be vertex H-pancyclic if every vertex of G is contained in an H-cycle of length l for every l in {3, . . ., |V (G)|}.A properly colored cycle in an edge-colored graph is a particular case of H-cycles in H-colored graph, namely when H is a complete graph with no loops.In this paper, we show sufficient conditions on an H-colored complete graph G to be vertex H-pancyclic.As a consequence, we obtain a well-known result about properly vertex pancyclicism in edge-colored complete graphs.