  Let $G$ be an edge-colored graph, a walk in $G$ is said to be a properly
colored walk iff each pair of consecutive edges have different colors,
including the first and the last edges in case that the walk be closed. Let $H$
be a graph possible with loops. We will say that a graph $G$ is an $H$-colored
graph iff there exists a function $c:E(G)\longrightarrow V(H)$. A path
$(v_1,\cdots,v_k)$ in $G$ is an $H$-path whenever $(c(v_1v_2),\cdots,$
$c(v_{k-1}v_k))$ is a walk in $H$, in particular, a cycle
$(v_1,\cdots,v_k,v_1)$ is an $H$-cycle iff $(c(v_1 v_2),\cdots,c(v_{k-1}v_k),$
$c(v_kv_1), c(v_1 v_2))$ is a walk in $H$. Hence, $H$ decide which color
transitions are allowed in a walk, in order to be an $H$-walk. Whenever $H$ is
a complete graph without loops, an $H$-walk is a properly colored walk, so
$H$-walk is a more general concept. In this paper, we work with $H$-colored
complete graphs, with restrictions given by an auxiliary graph. The main
theorems give conditions implying that every vertex in an $H$-colored complete
graph, is contained in an $H$-cycle of length 3 and in an $H$-cycle of length
4. As a consequence of the main results, we obtain some well-known theorems in
the theory of properly colored walks.
