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Let $G$ be an edge-colored graph, a walk in $G$ is said to be a properlycolored 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 $H$-coloredgraph 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 colortransitions are allowed in a walk, in order to be an $H$-walk. Whenever $H$ isa 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 $H$-coloredcomplete graphs, with restrictions given by an auxiliary graph. The maintheorems give conditions implying that every vertex in an $H$-colored completegraph, is contained in an $H$-cycle of length 3 and in an $H$-cycle of length4. As a consequence of the main results, we obtain some well-known theorems inthe theory of properly colored walks.

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