论文标题
有限$ 3 $ - 连接的合并本地$ 2 \ k_n $图形和$ S $ -ARC-TRANSTIVE图
Finite $3$-connected-set-homogeneous locally $2\K_n$ graphs and $s$-arc-transitive graphs
论文作者
论文摘要
在本文中，所有图形都被认为是有限的。 For $s\geq 1$ and a graph $\G$, if for every pair of isomorphic connected induced subgraphs on at most $s$ vertices there exists an automorphism of $\G$ mapping the first to the second, then we say that $\G$ is $s$-connected-set-homogeneous, and if every isomorphism between two isomorphic connected induced subgraphs on at大多数$ s $顶点可以扩展到$ \ g $的自动形态，然后我们说$ \ g $是$ s $ s $连接的均质。对于$ n \ geq 1 $，如果$ 2 \ k_n $的图形$ \ g $如果在$ \ g $的一组$ \ g $的顶点引起的子图$ [\ g（u）] $相邻的$ 2 \ k_n $。 请注意，$ 2 $连接的合成性但不是$ 2 $连接的均匀图只是半弧传递的图表，在代数图理论中是一个非常活跃的主题。在此激励的基础上，我们提出了描述或分类$ 3 $连接的围墙$ 3 $的$ 3 $连接的图表，这些图$ 3 $，不是$ 3 $连接的均匀性（EUR。J。Combin。93（2021）103275）。到目前为止，只有两个已知的家族有$ 3 $连接的环境图3 $的$ 3 $，不是$ 3 $连接的同质性，这些图形是本地的$ 2 \ k_n $，$ n = 2 $或$ 4 $。在本文中，我们完成了有限$ 3 $连接的固定图的分类，这些图是本地$ 2 \ k_n $的$ n \ geq 2 $，所有这些图形都是某些特定$ 2 $ -2 $ -Arc-arc-transansitive图的线图。此外，我们很好地描述了有限$ 3 $连接的合并，但不是$ 3 $连接的同质图，这些图是本地$ 2 \ k_n $，并且具有可解决的自动形态群体。然后，它用于构建一些新的$ 3 $连接的合并性，但不是$ 3 $连接的均匀图以及一些新的$ 2 $ -2 $ -ARC传播图。
In this paper, all graphs are assumed to be finite. For $s\geq 1$ and a graph $\G$, if for every pair of isomorphic connected induced subgraphs on at most $s$ vertices there exists an automorphism of $\G$ mapping the first to the second, then we say that $\G$ is $s$-connected-set-homogeneous, and if every isomorphism between two isomorphic connected induced subgraphs on at most $s$ vertices can be extended to an automorphism of $\G$, then we say that $\G$ is $s$-connected-homogeneous. For $n\geq 1$, a graph $\G$ is said to be locally $2\K_n$ if the subgraph $[\G(u)]$ induced on the set of vertices of $\G$ adjacent to a given vertex $u$ is isomorphic to $2\K_n$. Note that $2$-connected-set-homogeneous but not $2$-connected-homogeneous graphs are just the half-arc-transitive graphs which are a quite active topic in algebraic graph theory. Motivated by this, we posed the problem of characterizing or classifying $3$-connected-set-homogeneous graphs of girth $3$ which are not $3$-connected-homogeneous in (Eur. J. Combin. 93 (2021) 103275). Until now, there have been only two known families of $3$-connected-set-homogeneous graphs of girth $3$ which are not $3$-connected-homogeneous, and these graphs are locally $2\K_n$ with $n=2$ or $4$. In this paper, we complete the classification of finite $3$-connected-set-homogeneous graphs which are locally $2\K_n$ with $n\geq 2$, and all such graphs are line graphs of some specific $2$-arc-transitive graphs. Furthermore, we give a good description of finite $3$-connected-set-homogeneous but not $3$-connected-homogeneous graphs which are locally $2\K_n$ and have solvable automorphism groups. This is then used to construct some new $3$-connected-set-homogeneous but not $3$-connected-homogeneous graphs as well as some new $2$-arc-transitive graphs.
