The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study for GI and DGI-groups, defined analogously to the concept of CI and DCI-groups.
Tournaments are graphs obtained by assigning a direction for every edge in an undirected complete graph. We give a formula for the number of isomorphism classes of vertex-transitive tournaments with prime order. For that, we introduce Cayley tournaments, which are special Cayley digraphs, and show that there is a one to one identification between the isomorphism classes of vertex-transitive tournaments with prime order and the isomorphism classes of Cayley tournaments with prime order.
We study groups acting vertex-transitively on connected, trivalent graphs such that stabilizers of vertices are infinite. If the action is edge-transitive, we prove that the graph has to be a tree. We analyze the case where the action is not edge-transitive and fully classify the possible $2$-ended graphs. We draw connections to Willis' scale function and re-prove a result by Trofimov.
We observe that a ball of radius $1$ in the grandfather graph can not be realized as a ball of radius $1$ in a finite vertex transitive graph. We remark on when a ball in a finite vertex transitive graph appears as a ball in an infinite vertex transitive graph.
A graph is called cubic and tetravalent if all of its vertices have valency 3 and 4, respectively. It is called vertex-transitive and arc-transitive if its automorphism group acts transitively on its vertex-set and on its arc- set, respectively. In this paper, we combine some new theoretical results with computer calculations to construct all cubic vertex-transitive graphs of order at most 1280. In the process, we also construct all tetravalent arc-transitive graphs of order at most 640.
It is shown that every connected vertex-transitive graph of order $6p$, where $p$ is a prime, contains a Hamilton path. Moreover, it is shown that, except for the truncation of the Petersen graph, every connected vertex-transitive graph of order $6p$ which is not genuinely imprimitive contains a Hamilton cycle.
It is shown that every connected vertex-transitive graph of order $4p$, where $p$ is a prime, is hamiltonian with the exception of the Coxeter graph which is known to possess a Hamilton path.
Let $p$ be a prime and let $L$ be either the intransitive permutation group $C_p\times C_p$ of degree $2p$ or the transitive permutation group $C_p \wr C_2$ of degree $2p$. Let $Γ$ be a connected $G$-vertex-transitive and $G$-edge-transitive graph and let $v$ be a vertex of $Γ$. We show that if the permutation group induced by the vertex-stabiliser $G_v$ on the neighbourhood $Γ(v)$ is isomorphic to $L$ then either $|V(Γ)|\geq p|G_v|\log_p\left(|G_v|/2\right)$, or $|V(Γ)|$ is bounded by a constant depending only on $p$, or $Γ$ is a very-well understood graph. This generalises a few recent resul…
In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph $G(10,2)$, occurring as the graph $40$ in the Foster census of connected symmetric trivalent graphs.
I describe a 27-vertex graph that is vertex-transitive and edge-transitive but not 1-transitive. Thus while all vertices and edges of this graph are similar, there are no edge-reversing automorphisms.
A fascinating problem on digraphs is the existence problem of the finiteupper bound on s for all vertex-primitive s-arc-transitive digraphs except directed cycles (which is known to be reduced to the almost simple groups case). In this paper,we prove that s 2 for all G-vertex-primitive s-arc-transitive digraphs with G an (insoluble) alternating or symmetric group, which makes an important progress towards a solution of the problem. The proofs involves some methods that may be used to investigate other almost simple groups cases.
The main result of this paper is that, if $Γ$ is a finite connected $4$-valent arc-transitive graph, then either $Γ$ is part of a well-understood family of graphs, or every non-identity automorphism of $Γ$ fixes at most $1/3$ of the vertices. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs. Based on these results we propose a conjecture on the number of fixed points of non-identity automorphisms of vertex-transitive graphs of bounded valency.
We consider a finite, connected and simple graph $Γ$ that admits a vertex-transitive group of automorphisms $G$. Under the assumption that, for all $x \in V(Γ)$, the local action $G_x^{Γ(x)}$ is the action of $\mathrm{Sym}(n)$ on ordered pairs, we show that the group $G_x^{[3]}$, the pointwise stabiliser of a ball of radius three around $x$, is trivial.
Fakulteta za Matematiko in Fiziko, Univerza v Ljubljani（卢布尔雅那大学数学与物理学院）；Mathematics Institute, University of Warwick（华威大学数学研究所）；Dipartimento di Matematica e Applicazioni, University of Milano-Bicocca（米兰比可卡大学数学与应用系）
A graph is said to be {\em vertex-transitive non-Cayley} if its full automorphism group acts transitively on its vertices and contains no subgroups acting regularly on its vertices. In this paper, a complete classification of cubic vertex-transitive non-Cayley graphs of order $12p$, where $p$ is a prime, is given. As a result, there are $11$ sporadic and one infinite family of such graphs, of which the sporadic ones occur when $p=5$, $7$ or $17$, and the infinite family exists if and only if $p\equiv1\ (\mod 4)$, and in this family there is a unique graph for a given order.
In this paper we discuss a method for bounding the size of the stabiliser of a vertex in a $G$-vertex-transitive graph $Γ$. In the main result the group $G$ is quasiprimitive or biquasiprimitive on the vertices of $Γ$, and we obtain a genuine reduction to the case where $G$ is a nonabelian simple group. Using normal quotient techniques developed by the first author, the main theorem applies to general $G$-vertex-transitive graphs which are $G$-locally primitive (respectively, $G$-locally quasiprimitive), that is, the stabiliser $G_α$ of a vertex $α$ acts primitively (respectively quasiprimitiv…
A famous conjecture of Lovász states that every connected vertex-transitive graph contains a Hamilton path. In this article we confirm the conjecture in the case that the graph is dense and sufficiently large. In fact, we show that such graphs contain a Hamilton cycle and moreover we provide a polynomial time algorithm for finding such a cycle.
Let $k$ be an integer. We prove a rough structure theorem for separations of order at most $k$ in finite and infinite vertex transitive graphs. Let $G = (V,E)$ be a vertex transitive graph, let $A \subseteq V$ be a finite vertex-set with $|A| \le |V|/2$ and $|\{v \in V \setminus A : {$u \sim v$ for some $u \in A$} \}|\le k$. We show that whenever the diameter of $G$ is at least $31(k+1)^2$, either $|A| \le 2k^3+k^2$, or $G$ has a ring-like structure (with bounded parameters), and $A$ is efficiently contained in an interval. This theorem may be viewed as a rough characterization, generalizing a…
We prove that, given a finite graph $Σ$ satisfying some mild conditions, there exist infinitely many tetravalent half-arc-transitive normal covers of $Σ$. Applying this result, we establish the existence of infinite families of finite tetravalent half-arc-transitive graphs with certain vertex stabilizers, and classify the vertex stabilizers up to order $2^8$ of finite connected tetravalent half-arc-transitive graphs. This sheds some new light on the longstanding problem of classifying the vertex stabilizers of finite tetravalent half-arc-transitive graphs.
In this paper we investigate orders, longest cycles and the number of cycles of automorphisms of finite vertex-transitive graphs. In particular, we show that the order of every automorphism of a connected vertex-transitive graph with $n$ vertices and of valence $d$, $d\le 4$, is at most $c_d n$ where $c_3=1$ and $c_4 = 9$. Whether such a constant $c_d$ exists for valencies larger than $4$ remains an unanswered question. Further, we prove that every automorphism $g$ of a finite connected $3$-valent vertex-transitive graph $Γ$, $Γ\not\cong K_{3,3}$, has a regular orbit, that is, an orbit of $\la…
In this paper we study $G$-arc-transitive graphs $Δ$ where the permutation group $G_x^{Δ(x)}$ induced by the stabiliser $G_x$ of the vertex $x$ on the neighbourhood $Δ(x)$ satisfies the two conditions given in the introduction. We show that for such a $G$-arc-transitive graph $Δ$, if $(x,y)$ is an arc of $Δ$, then the subgroup $G_{x,y}^{[1]}$ of $G$ fixing pointwise $Δ(x)$ and $Δ(y)$ is a $p$-group for some prime $p$. Next we prove that every $G$-locally primitive (respectively quasiprimitive, semiprimitive) graph satisfies our two local hypotheses. Thus this provides a new Thompson-Wielandt-l…
The main result of this paper is that, if $Γ$ is a connected 4-valent $G$-arc-transitive graph and $v$ is a vertex of $Γ$, then either $Γ$ is one of a well understood infinite family of graphs, or $|G_v|\leq 2^43^6$ or $2|G_v|\log_2(|G_v|/2)\leq |\VΓ|$ and that this last bound is tight. As a corollary, we get a similar result for $3$-valent vertex-transitive graphs.
We study vertex-quasiprimitive $2$-arc-transitive digraphs, reducing the problem of vertex-primitive $2$-arc-transitive digraphs to almost simple groups. This includes a complete classification of vertex-quasiprimitive $2$-arc-transitive digraphs where the action on vertices has O'Nan-Scott type SD or CD.
This paper gives lower bounds on the spectral radius of vertex-transitive graphs, based on the number of ``prime cycles'' at a vertex. The bounds are obtained by constructing circuits in the graph that resemble ``cactus trees'', and enumerating them. Counting these circuits gives a coefficient-wise underestimation of the Green function of the graph, and hence and underestimation of its spectral radius. The bounds obtained are very good for the Cayley graph of surface groups of genus g>=2, with standard generators (these graphs are the 1-skeletons of tessellations of hyperbolic plane by 4g-gons…
We introduce the concept of the primitivity of independent set in vertex-transitive graphs, and investigate the relationship between the primitivity and the structure of maximum independent sets in direct products of vertex-transitive graphs. As a consequence of our main results, we positively solve an open problem related to the structure of independent sets in powers of vertex-transitive graphs.
A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and vertex transitive graphs with cores half their size always admit such partitions. We also show that the vertex sets of vertex transitive graphs with cores less than half their size do not, in general, ha…
We introduce uniformly vertex-transitive graphs as vertex-transitive graphs satisfying a stronger condition on their automorphism groups, motivated by a problem which arises from a Sinkhorn-type algorithm. We use the derangement graph $D(Γ)$ of a given graph $Γ$ to show that the uniform vertex-transitivity of $Γ$ is equivalent to the existence of cliques of sufficient size in $D(Γ)$. Using this method, we find examples of graphs that are vertex-transitive but not uniformly vertex-transitive, settling a previously open question. Furthermore, we develop sufficient criteria for uniform vertex-tra…
We prove that every regular graph of valency at least four whose automorphism group contains a nilpotent subgroup acting transitively on the vertex set admits a nowhere-zero 3-flow.
We study $G$-vertex-primitive and $(G,s)$-arc-transitive digraphs for almost simple groups $G$ with socle $\mathrm{PSL}_n(q)$. It turns out that $s\leqslant2$ for such digraphs, which provides the first step in determining an upper bound on $s$ for all the vertex-primitive $s$-arc-transitive digraphs.
In 2017, Giudici, Li and the third author constructed the first known family of vertex-primitive $2$-arc-transitive digraphs of valency at least $2$. The smallest digraph in this family admits $\mathrm{PSL}_3(49)$ acting $2$-arc-transitively with vertex-stabilizer $\mathrm{A}_6$ and hence has $30758154560$ vertices. In this paper, we prove that this digraph is the vertex-primitive $2$-arc-transitive digraph of valency at least $2$ with fewest vertices.
We describe two similar but independently-coded computations used to construct a complete catalogue of the transitive groups of degree less than $48$, thereby verifying, unifying and extending the catalogues previously available. From this list, we construct all the vertex-transitive graphs of order less than $48$. We then present a variety of summary data regarding the transitive groups and vertex-transitive graphs, focussing on properties that seem to occur most frequently in the study of groups acting on graphs. We illustrate how such catalogues can be used, first by finding a complete list…
For any finite, undirected, non-bipartite, vertex-transitive graph, we establish an explicit lower bound for the smallest eigenvalue of its normalised adjacency operator, which depends on the graph only through its degree and its isoperimetric constant.
We prove bounds on the chromatic number $χ$ of a vertex-transitive graph in terms of its clique number $ω$ and maximum degree $Δ$. We conjecture that every vertex-transitive graph satisfies $χ\le \max \left\{ω, \left\lceil\frac{5Δ+ 3}{6}\right\rceil\right\}$ and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with $Δ\ge 13$ we prove the Borodin-Kostochka conjecture, i.e., $χ\le\max\{ω,Δ-1\}$.
It is well-known that concentrators are sparse graphs of high connectivity, which play a key role in the construction of switching networks; and any semi-vertex transitive graph is isomorphic to a bi-coset graph. In this paper, we prove that random bi-coset graphs are almost always concentrators, and construct some examples of semi-vertex transitive concentrators.
We investigate the structure of a digraph having a transitive automorphism group where every cutset of minimal cardinality consists of all successors or all predecessors of some vertex. We improve most of the existing results in this area.
A clique (resp., independent set) in a graph is strong if it intersects every maximal independent sets (resp., every maximal cliques). A graph is CIS if all of its maximal cliques are strong and localizable if it admits a partition of its vertex set into strong cliques. In this paper we prove that a clique $C$ in a vertex-transitive graph $Γ$ is strong if and only if $|C||I|=|V(Γ)|$ for every maximal independent set $I$ of $Γ$. Based on this result we prove that a vertex-transitive graph is CIS if and only if it admits a strong clique and a strong independent set. We classify all vertex-transi…
We generalise the standard constructions of a Cayley graph in terms of a group presentation by allowing some vertices to obey different relators than others. The resulting notion of presentation allows us to represent every vertex transitive graph. As an intermediate step, we prove that every countably infinite, connected, vertex transitive graph has a perfect matching. Incidentally, we construct an example of a 2-ended cubic vertex transitive graph which is not a Cayley graph, answering a question of Watkins from 1990.
In 1979 Babai found a clever argument to prove that every connected vertex transitive graph on $n \ge 3$ vertices contains a cycle of length at least $\sqrt{3n}$. Here we modify his approach to show that such graphs must contain a cycle of length at least $(1 - o(1))n^{3/5}$.
It has long been known that there exist finite connected tetravalent arc-transitive graphs with arbitrarily large vertex-stabilisers. However, beside a well known family of exceptional graphs, related to the lexicographic product of a cycle with an edgeless graph on two vertices, only a few such infinite families of graphs are known. In this paper, we present two more families of tetravalent arc-transitive graphs with large vertex-stabilisers, each significant for its own reason.
For a non-complete graph $Γ$, a vertex triple $(u,v,w)$ with $v$ adjacent to both $u$ and $w$ is called a $2$-geodesic if $u\neq w$ and $u,w$ are not adjacent. Then $Γ$ is said to be $2$-geodesic transitive if its automorphism group is transitive on both arcs and 2-geodesics. In previous work the author showed that if a $2$-geodesic transitive graph $Γ$ is locally disconnected and its automorphism group $\Aut(Γ)$ has a non-trivial normal subgroup which is intransitive on the vertex set of $Γ$, then $Γ$ is a cover of a smaller 2-geodesic transitive graph. Thus the `basic' graphs to study are th…
A map $X$ on a surface is called vertex-transitive if the automorphism group of $X$ acts transitively on the set of vertices of $X$. If the face-cycles at all the vertices in a map are of same type then the map is called semi-equivelar. In general, semi-equivelar maps on a surface form a bigger class than vertex-transitive maps. There are semi-equivelar toroidal maps which are not vertex-transitive. In this article, we show that semi-equivelar toroidal maps are quotients of vertex-transitive toroidal maps. More explicitly, we prove that each semi-equivelar toroidal map has a finite vertex-tran…
A connected graph whose automorphism group acts transitively on the edges and vertices, but not on the set of ordered pairs of adjacent vertices of the graph is called half-arc-transitive. It is well known that the valence of a half-arc-transitive graph is even and at least four. Several infinite families of half-arc-transitive graphs of valence four are known, however, in all except four of the known specimens, the vertex-stabiliser in the automorphism group is abelian. The first example of a half-arc-transitive graph of valence four and with a non-abelian vertex-stabiliser was described in […
A graph of order $n$ is $p$-factor-critical, where $p$ is an integer of the same parity as $n$, if the removal of any set of $p$ vertices results in a graph with a perfect matching. 1-Factor-critical graphs and 2-factor-critical graphs are factor-critical graphs and bicritical graphs, respectively. It is well known that every connected vertex-transitive graph of odd order is factor-critical and every connected non-bipartite vertex-transitive graph of even order is bicritical. In this paper, we show that a simple connected vertex-transitive graph of odd order at least 5 is 3-factor-critical if …
The distinguishing number of a graph $G$ is the smallest $k$ such that $G$ admits a $k$-colouring for which the only colour-preserving automorphism of $G$ is the identity. We determine the distinguishing number of finite $4$-valent vertex-transitive graphs. We show that, apart from one infinite family and finitely many examples, they all have distinguishing number $2$.
Quasi-vertex-transitive maps are the homogeneous maps on the plane with finitely many vertex orbits under the action of their automorphism groups. We show that there exist quasi-vertex-transitive maps of types $[p^3, 3]$ for $p \equiv 1$ (mod $6$), but there doesn't exist vertex-transitive map of such types. In particular, we determine the surface with the lowest possible genus that admit a polyhedral map of type $[5^3, 3]$.
It is shown that there are infinitely many connected vertex-transitive graphs that have no Hamilton decomposition, including infinitely many Cayley graphs of valency 6, and including Cayley graphs of arbitrarily large valency.
A step forward is made in a long standing Lovász's problem regarding hamiltonicity of vertex-transitive graphs by showing that every connected vertex-transitive graph of order a product of two primes, other than the Petersen graph, contains a Hamilton cycle. Essential tools used in the proof range from classical results on existence of Hamilton cycles, such as Chvátal's theorem and Jackson's theorem, to certain results on polynomial representations of quadratic residues at primitive roots in finite fields.
