Dynamic cycles in edge-colored multigraphs

Abstract

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 $\{0,\ldots,n-1\}$, $k_i \geq 1$ 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 $v_i\neq v_j$, 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 \textit{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 $c\geq 3$ colors.