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arXiv 2303.02548math.CO
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边着色重图中的动态圈

Dynamic cycles in edge-colored multigraphs

Hortensia Galeana-Sánchez, Carlos Vilchis-Alfaro

备注：
arXiv admin note: text overlap with arXiv:2207.03623
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摘要：

设$H$是一个可能有环的图，$G$是一个无环的重图. $G$的$H$ -着色是一个函数$c: E(G) \rightarrow V(H)$。当我们对$G$进行固定的$H$ -染色时，我们称$G$是$H$ -染色的重图.具有端顶点$u$和$v$的所有边的集合将由$E_{uv}$表示。我们将说$W=(v_0,e_0^1, \ldots, e_0^{k_0},v_1,e_1^1,\ldots,e_1^{k_1},v_2,\ldots,v_{n-1},e_{n-1}^1,\ldots,e_{n-1}^{k_{n-1}},v_n)$，其中对于$\{0,\ldots,n-1\}$中的每个$i$，对于每个$j \in \{1,\ldots, k_i \}$，$k_i \geq 1$和$e_i^j \in E_{v_iv_{i+1}}$是动态$H$ -游动当且仅当对于每个$i \in \{0,\ldots,n-2\}$，$c(e_i^{k_i})c(e_{i+1}^1)$是$H$中的边.只要$v_0=v_n$和$c(e_{n-1}^{k_{n-1}})c(e_0^1)$是$H$中的边，我们就说动态$H$-游动是封闭的动态$H$ -游动。此外，对于每个$\{i,j\}\subseteq \{0,\ldots,v_{n-1}\}$，当$v_i\neq v_j$时，封闭的动态$H$ -行走被称为动态$H$ -循环。特别地，动态$H$ -游走是一个$H$ -游走，只要$k_i=1$，对于每个$i \in \{0,\ldots,n-1\}$，并且当$H$是一个没有循环的完全图时，$H$ -游走被称为适当着色的游走.本文研究了$H$着色重图中动态$H$ -圈、动态$H$ -迹和动态$H$ -路的存在性和长度.为了实现这一点，我们引入了一个新的概念的颜色程度，即\textit{dynamic degree}，这使我们能够扩展一些经典的结果，如Ore的定理，$H$着色的多重图。同时，我们给出了$H$ -着色重图中存在Hamilton动态$H$ -圈的充分条件，并由此得到了边着色重图中存在适当着色Hamilton圈的充分条件，且至少有$c\geq 3$色.

英文摘要：

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.

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