论文标题
无限程度的图形的区分指数
The distinguishing index of graphs with infinite minimum degree
论文作者
论文摘要
图$ g $的区分索引$ d'（g）$是获得$ g $的边缘着色所需的最小颜色，仅由琐碎的自动形态保留。我们表明，如果$ g $是连接的$α$ -Regrumar图，对于某些无限的红衣主教$α$，那么$ d'（g）\ le 2 $，证明了Lehner，Pilśniak和Stawiski的猜想。我们还表明，如果$ g $是一个具有无限度的图表，最多只能为每个无限的红衣主教$α$ $ 2^α$ a $α$的顶点$α$，则是$ d'（g）\ le 3 $。特别是，$ d'（g）\ le 3 $如果$ g $具有无限的最低学位和订单，最多$ 2^{\ aleph_0} $。
The distinguishing index $D'(G)$ of a graph $G$ is the least number of colors necessary to obtain an edge coloring of $G$ that is preserved only by the trivial automorphism. We show that if $G$ is a connected $α$-regular graph for some infinite cardinal $α$ then $D'(G) \le 2$, proving a conjecture of Lehner, Pilśniak, and Stawiski. We also show that if $G$ is a graph with infinite minimum degree and at most $2^α$ vertices of degree $α$ for every infinite cardinal $α$, then $D'(G) \le 3$. In particular, $D'(G) \le 3$ if $G$ has infinite minimum degree and order at most $2^{\aleph_0}$.
