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

Abstract

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.