Let H be a graph possibly with loops and G be a multigraph without loops. An H-coloring of G is a function \(c: E(G) \rightarrow V(H)\) . We will say that G is an H-colored multigraph, whenever we are taking a fixed H-coloring of G. The set of all the edges with end vertices u and v will be denoted by \(E_{uv}\) . We will say that \(W=(v_0,e_0^1, \ldots , e_0^{k_0},v_1,e_1^1,\ldots ,\) \(e_1^{k_1},v_2,\ldots ,v_{n-1},e_{n-1}^1,\ldots ,e_{n-1}^{k_{n-1}},v_n)\) , where for each i in \(\{0,\ldots ,\) \(n-1\}\) , \(k_i \ge 1\) and \(e_i^j \in E_{v_iv_{i+1}}\) for every \(j \in \{1,\ldots , k_i \}\) , is a dynamic H-walk iff \(c(e_i^{k_i})c(e_{i+1}^1)\) is an edge in H, for each \(i \in \{0,\ldots ,n-2\}\) . We will say that a dynamic H-walk is a closed dynamic H-walk whenever \(v_0=v_n\) and \(c(e_{n-1}^{k_{n-1}})c(e_0^1)\) is an edge in H. Moreover, a closed dynamic H-walk is called dynamic H-cycle whenever \(v_i\ne v_j\) , for every \(\{i,j\}\subseteq \{0,\ldots ,v_{n-1}\}\) . In particular, a dynamic H-walk is an H-walk whenever \(k_i=1\) , for every \(i \in \{0,\ldots ,n-1\}\) , and when H is a complete graph without loops, an H-walk is well known as a properly colored walk. In this work, we study the existence and length of dynamic H-cycles, dynamic H-trails and dynamic H-paths in H-colored multigraphs. To accomplish this, we introduce a new concept of color degree, namely, the dynamic degree, which allows us to extend some classic results, as Ore’s Theorem, for H-colored multigraphs. Also, we give sufficient conditions for the existence of hamiltonian dynamic H-cycles in H-colored multigraphs, and as a consequence, we obtain sufficient conditions for the existence of properly colored hamiltonian cycle in edge-colored multigraphs, with at least \(c\ge 3\) colors.
