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 \(c:E(G)\longrightarrow V(H)\) . A cycle \((v_1,\ldots ,v_k,v_1)\) is an H-cycle if and only if \((c(v_1 v_2),\ldots ,c(v_{k-1}v_k),\) \(c(v_kv_1), c(v_1 v_2))\) 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.
