Finite 3-set-homogeneous graphs

Abstract

In this paper, all graphs are assumed to be finite. Let $s\ge 1$ be an integer. A graph is called $s$-CSH ($s$-connected-set-homogeneous) if for every pair of isomorphic connected induced subgraphs on at most $s$ vertices, there exists an automorphism mapping the first to the second. A graph is called $s$-SH ($s$-set-homogeneous) if for every pair of isomorphic induced subgraphs (not necessarily connected) on at most $s$ vertices, there exists an automorphism mapping the first to the second. A graph is called $s$-homogeneous (respectively $s$-CH, that is, $s$-connected-homogeneous) if every isomorphism between two induced subgraphs (respectively, connected induced subgraphs) on at most $s$ vertices extends to an automorphism of the whole graph.

The first main result, Theorem 1.1, proves that each connected 3-CSH graph is arc-transitive. A consequence of this result is that each 3-CSH graph is 2-CH. Note that 2-CSH but not 2-CH graphs are just half-arc-transitive graphs which have been extensively studied in the literature. Motivated by this, it is natural to consider 3-CSH but not 3-CH graphs. In this paper, we first prove that there exist infinitely many 3-CSH but not 3-CH graphs, and then prove that every prime valent 3-CSH graph is 3-CH. Finally, using these two results, we classify all arc-regular 3-CSH but not 3-CH graphs of girth 3.

Introduction

All groups considered in this paper are finite, and all graphs are finite, connected, simple and undirected, unless explicitly stated. For group-theoretic and graph-theoretic terminology not defined here we refer the reader to [3], [44].

A graph is called (respectively connected) homogeneous if any isomorphism between any two isomorphic (respectively connected) induced subgraphs extends to an automorphism of the graph. Homogeneity is the strongest possible symmetry property that a graph can have. The finite (connected) homogeneous graphs have been classified [18], [19], and very few families of graphs arise. If we consider only certain type of subgraphs, we can have various relaxations of homogeneity. For example, $s$-arc-transitivity and $s$-geodesic-transitivity are two important variations.

Let $\Gamma =(V,E)$ be a graph with vertex set $V$ and edge set $E$. We use $\text{Aut}\left(\Gamma \right)$ to denote the full automorphism group of $\Gamma$. An $s$-arc, $s\ge 1$, in $\Gamma$ is an ordered $(s+1)$-tuple $(v_0,v_1,\dots ,v_{s-1},v_s)$ of vertices of $\Gamma$ such that $v_{i-1}$ is adjacent to $v_i$ for $1\le i\le s$, and $v_{i-1}\ne v_{i+1}$ for $1\le i\le s-1$, and a $1$-arc is usually called an arc. Let $G\le \text{Aut}\left(\Gamma \right)$. A graph $\Gamma$ is said to be $(G,s)$-arc-transitive if $\Gamma$ has at least one $s$-arc and $G$ is transitive on the set of $s$-arcs of $\Gamma$. A $(G,s)$-arc-transitive graph is sometimes simply called an $s$-arc-transitive graph. A fair amount of work has been done on $s$-arc-transitive graphs. In [40], [41], Tutte proved that $s$-arc transitive graphs of valency three satisfy $s\le 5$, and this was generalized by Weiss [43] who proved that if the valency is at least three, then $s\le 7$. Following these two works, $s$-arc-transitive graphs have been extensively studied over decades by many authors. For more results about $s$-arc-transitive graphs, see, for example, [17], [21], [28], [36], [37], [42]. An $s$-arc $(v_0,v_1,\dots ,v_{s-1},v_s)$ of $\Gamma$ is called an $s$-geodesic if $v_0$ and $v_s$ are at distance $s$. A graph $\Gamma$ is said to be $s$-geodesic-transitive if $\Gamma$ has at least one $s$-geodesic and $\text{Aut}\left(\Gamma \right)$ is transitive on the set of $s$-geodesics of $\Gamma$. For a graph with girth at least $s+1$, $s$-geodesic transitivity is the same as $s$-arc-transitivity. (The girth of a graph $\Gamma$ is the length of a shortest cycle of $\Gamma$.) Recently, some beautiful works were done on $2$-geodesic transitive graphs (see [11], [12], [13], [14], [15]).

Stronger than the above two symmetry conditions we have $s$-homogeneity. Let $s$ be a positive integer and let $\Gamma$ be a graph. For a subgroup $G$ of $\text{Aut}\left(\Gamma \right)$, we say that $\Gamma$ is $(G,s)$-homogeneous (respectively $(G,s)$-CH, that is, $(G,s)$-connected-homogeneous) if every isomorphism between two isomorphic induced subgraphs (respectively connected induced subgraphs) on at most $s$ vertices can be extended to an automorphism, say $g$, of $\Gamma$ so that $g\in G$, in which case, $\Gamma$ is sometimes called $s$-homogeneous (respectively $s$-CH, that is, $s$-connected-homogeneous). Clearly, a graph is $1$-homogeneous if and only if it is vertex-transitive. A graph $\Gamma$ is 2-homogeneous if and only if $\Gamma$ and its complement are both arc-transitive. This implies that a connected 2-homogeneous graph is either a complete multipartite graph or an orbital graph of a primitive permutation group of rank 3. Primitive permutation groups of rank 3 have been classified in [2], [27], [32], [33]. Consequently, 2-homogeneous graphs are in some sense known. For $s$-CH graphs, it is easy to see that a graph is $2$-CH if and only if it is 1-arc-transitive, and a graph of girth at least $4$ is $3$-CH if and only if it is $2$-arc-transitive. Infinite $3$-CH graphs with more than one end were classified in [25], and finite $k$-CH graphs with $k\ge 3$ were investigated in [1], [10], [31].

Another symmetry condition weaker than $s$-homogeneity is $s$-set-homogeneity. Let $s$ be a positive integer and let $\Gamma$ be a graph. For a subgroup $G$ of $\text{Aut}\left(\Gamma \right)$, we say that $\Gamma$ is $(G,s)$-CSH ($(G,s)$-connected-set-homogeneous) if for any pair of isomorphic connected induced subgraphs of $\Gamma$ on at most $s$ vertices there exists $g\in G$ mapping the first to the second, and that $\Gamma$ is $(G,s)$-SH ($(G,s)$-set-homogeneous) if for any pair of isomorphic induced subgraphs (not necessarily connected) of $\Gamma$ on at most $s$ vertices there exists $g\in G$ mapping the first to the second. A $(G,s)$-CSH (respectively $(G,s)$-SH) graph is also simply called an $s$-SCH (respectively $s$-SH) graph. A $1$-SH graph is vertex-transitive and a $2$-CSH graph is vertex- and edge-transitive. It is natural to study $3$-CSH graphs. In [25], the infinite $3$-CSH graphs with more than one end were classified. In this paper, we shall be concerned with finite $3$-SH and $3$-CSH graphs. Our first result is the following theorem.

Theorem 1.1

Let $\Gamma$ be a connected $(G,3)$-CSH graph with $G\le \text{Aut}\left(\Gamma \right)$. Then either $\Gamma$ is $G$-arc-transitive or $\Gamma$ is a triangle.

We observe that if a graph $\Gamma$ is $(G,3)$-SH with $G\le \text{Aut}\left(\Gamma \right)$ then its complement ${\Gamma }^c$ is also $(G,3)$-SH, and hence by the above theorem, ${\Gamma }^c$ is also $G$-arc-transitive. Consequently, we have the following corollary.

Corollary 1.2

Every $3$-SH graph is $2$-homogeneous.

Theorem 1.1 implies that every $3$-CSH graph is $2$-CH. Recall that a graph is said to be half-arc-transitive if it is vertex- and edge-transitive but not arc-transitive. Constructing and characterizing half-arc-transitive graphs is also currently an active topic in algebraic graph theory — see [8], [9], [34], [35], [38], [46] for example. By definition, a graph is half-arc-transitive if and only if it is $2$-CSH but not $2$-CH. Analogous to half-arc-transitive graphs, it would be interesting to consider $3$-CSH but not $3$-CH graphs.

Let $\Gamma$ be a $(G,3)$-CSH graph for some $G\le \text{Aut}\left(\Gamma \right)$. If $\Gamma$ has girth larger than $3$ then it is $(G,2)$-path-transitive, meaning that $G$ is transitive on the set of $2$-paths of $\Gamma$. The class of $2$-path-transitive graphs is larger than the class of $2$-arc-transitive graphs. In [30], the structure of the vertex-stabilizer and the amalgam for the full automorphism group of a $2$-path-transitive but not $2$-arc-transitive graph are determined, and in [29], a classification of vertex-primitive and vertex-biprimitive $2$-path-transitive graphs which are not $2$-arc-transitive is given. Motivated by these facts, we shall consider $3$-CSH but not $3$-CH graphs of girth $3$. The first natural question is:

Question A

Do there exist $3$-CSH but not $3$-CH graphs of girth $3$?

Before giving an answer to this question, we shall give some basic properties of a $(G,3)$-CSH graph. First we fix some notation. For a graph $\Gamma$ and $u\in V\left(\Gamma \right)$, ${\Gamma }_1\left(u\right)$ denotes the set of neighbors of $u$ in $\Gamma$. Use ${\Gamma }^c$ to denote the complementary graph of $\Gamma$. For $B\subseteq V\left(\Gamma \right)$, $\left[B\right]$ denotes the subgraph induced by $B$. Let $H$ be a transitive permutation group on a finite set $\Omega$. The number of orbits of $H_{\alpha }$ with $\alpha \in \Omega$ is called the rank of $H$.

Theorem 1.3

Let $\Gamma$ be a $(G,3)$-CSH non-complete graph of girth $3$ with $G\le \text{Aut}\left(\Gamma \right)$. Then for any $\{u,v\}\in E\left(\Gamma \right)$, we have the following:

• If $\left[{\Gamma }_1\left(u\right)\right]$ is connected, then $\left[{\Gamma }_1\left(u\right)\right]$ is of diameter $2$, and if $\left[{\Gamma }_1\left(u\right)\right]$ is disconnected, then $\left[{\Gamma }_1\left(u\right)\right]\cong m{K}_{\ell }$ for some positive integers $m,\ell$.

• $G_u$ is edge-transitive on ${\left[{\Gamma }_1\left(u\right)\right]}^c$.

• $G_{uv}$ has $s$ orbits on ${\Gamma }_1\left(u\right)\cap {\Gamma }_1\left(v\right)$ with equal size, where $s=1,2,3$ or $6$.

• $G_{uv}$ has $t$ orbits on ${\Gamma }_1\left(u\right)-({\Gamma }_1\left(u\right)\cap {\Gamma }_1\left(v\right))\cup \left\{v\right\}$ with equal size, where $t=1,2$.

• $G_u^{{\Gamma }_1\left(u\right)}$ has rank $r=1+s+t\text{(}=3,4,5,6,8$, or $9$).

For a $2$-arc $(v_0,v_1,v_2)$ of a graph $\Gamma$, $(v_2,v_1,v_0)$ is also a $2$-arc. If we identify these two arcs, then we obtain a $2$-path, denoted by $[v_0,v_1,v_2]$, and if $v_0$ and $v_2$ are adjacent then we get a triangle, denoted by $\{v_0,v_1,v_2\}$. The $2$-path $[v_0,v_1,v_2]$ is called a $2$-geodesic-path provided that the triple $(v_0,v_1,v_2)$ is a $2$-geodesic. As a corollary of the above theorem, we have the following result which is useful in constructing $3$-CSH graphs.

Corollary 1.4

With the notation as Theorem 1.3, we have the follows.

• If $s=1$, then $G$ is transitive on the set of $3$-tuples $(u_1,u_2,u_3)$ such that $\{u_1,u_2,u_3\}$ is a triangle, and if $t=1$, then $G$ is transitive on the set of $2$-geodesic-paths of $\Gamma$.

• If $s=2$, then $G_u$ is transitive on the edge-sets of both $\left[{\Gamma }_1\left(u\right)\right]$ and ${\left[{\Gamma }_1\left(u\right)\right]}^c$.

The next theorem answers Question A in positive. To state the result, we introduce some notation. We use ${\mathbb{Z}}_n$ to denote the cyclic group of order $n$. For two groups $M$ and $N$, we use $N⋊M$ to denote a semi-direct product of $N$ by $M$ (with kernel $N$ and complement $M$). Let $H$ and $G$ be permutation groups on sets $X$ and $Y$, respectively. The wreath product $H\wr G$ is the semi-direct product $H^{\left|Y\right|}⋊G$ where $G$ operates on $H^{\left|Y\right|}$ by permuting the components.

Theorem 1.5

• Let ${\Gamma }_{3,2}$ be a trivalent $2$-arc-transitive graph such that the edge stabilizer in $\text{Aut}\left({\Gamma }_{3,2}\right)$ is the cyclic group of order $4$. Then the line graph ${\Gamma }_{3,2}^{\prime }$ of ${\Gamma }_{3,2}$ is $3$-CSH but not $3$-CH.

• Suppose that ${\Gamma }_{5,2}$ is a pentavalent $2$-arc-transitive graph whose edge-stabilizer and vertex-stabilizer are isomorphic to ${\mathbb{Z}}_4\wr {\mathbb{Z}}_2$ and $F_{20}\times {\mathbb{Z}}_4$, respectively, where $F_{20}={\mathbb{Z}}_5⋊{\mathbb{Z}}_4$ is a Frobenius group. Then the line graph ${\Gamma }_{5,2}^{\prime }$ of ${\Gamma }_{5,2}$ is $3$-CSH but not $3$-CH.

Motivated by the facts above, we would like to pose the following problem.

Problem B

Characterize or classify $3$-CSH graphs of girth $3$ which are not $3$-CH.

Note that in [14] a classification of prime valent $3$-CH graphs of girth $3$ is given. Motivated by this, we prove the following theorem from which all prime valent $3$-CSH graphs of girth $3$ are known.

Theorem 1.6

Every prime valent $3$-CSH graph of girth $3$ is also $3$-CH.

Using this theorem, our next result gives a classification of arc-regular $3$-CSH but not $3$-CH graphs of girth $3$. Recall that a graph $\Gamma$ is called arc-regular if for any two arcs of $\Gamma$, there is a unique automorphism of $\Gamma$ that maps one to the other.

Theorem 1.7

An arc-regular graph of girth $3$ is $3$-CSH but not $3$-CH if and only if it is isomorphic to ${\Gamma }_{3,2}^{\prime }$ (as given in Theorem 1.5 (1)).