论文标题
总$ K $ - 均匀图
Total $k$-Uniform Graphs
论文作者
论文摘要
如果序列中的每个顶点$ v $都有一个邻居，该序列在序列中的每个顶点$ v $中都有一个邻居，而序列中没有顶点$ v $，并且最终每个$ g $的顶点在序列中至少一个邻居。总统治顺序的最小和最大长度是$ g $的总支配数（由$γ_T（g）$表示）和Grundy的总统治数为$ G $（分别由$γ_{gr}^t（g）$表示。在本文中，我们研究了总统治数量和Grundy总统治数的图。对于每个正整数$ k $，如果$ g $ a $ k $ - 均匀图，则如果$γ_t（g）=γ_{gr}^t（g）= k $。我们证明，当$ k $奇怪时，没有$ k $均匀的图。此外，我们提供了一个总4均匀图，该图作为T. Gologranc等人的猜想的反例。并提供连接的总8均匀图。此外，我们证明，每个$ k $均匀的，连接和虚假双胞胎的总均匀图都是每偶数$ k $的常规图。我们还表明，没有$ k $ khordal连接的图形，带有$ k \ geq 4 $，并且表征了所有总$ k $均匀的和弦图。
A sequence of vertices in a graph $G$ without isolated vertices is called a total dominating sequence if every vertex $v$ in the sequence has a neighbor which is adjacent to no vertex preceding $v$ in the sequence, and at the end every vertex of $G$ has at least one neighbor in the sequence. Minimum and maximum lengths of a total dominating sequence is the total domination number of $G$ (denoted by $γ_t(G)$) and the Grundy total domination number of $G$ (denoted by $γ_{gr}^t(G)$), respectively. In this paper, we study graphs with equal total domination number and Grundy total domination number. For every positive integer $k$, we call $G$ a total $k$-uniform graph if $γ_t(G)=γ_{gr}^t(G)=k$. We prove that there is no total $k$-uniform graph when $k$ is odd. In addition, we present a total 4-uniform graph which stands as a counterexample for a conjecture by T. Gologranc et.al. and provide a connected total 8-uniform graph. Moreover, we prove that every total $k$-uniform, connected and false twin-free graph is regular for every even $k$. We also show that there is no total $k$-uniform chordal connected graph with $k\geq 4$ and characterize all total $k$-uniform chordal graphs.
