139 research outputs found

    Arc-transitive graphs of valency 8 have a semiregular automorphism

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    One version of the polycirculant conjecture states that every vertex-transitive graph has a semiregular automorphism. We give a proof of the conjecture in the arc-transitive case for graphs of valency 8, which was the smallest open case.Comment: 5 page

    Semiregular automorphisms of edge-transitive graphs

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    The polycirculant conjecture asserts that every vertex-transitive digraph has a semiregular automorphism, that is, a nontrivial automorphism whose cycles all have the same length. In this paper we investigate the existence of semiregular automorphisms of edge-transitive graphs. In particular, we show that any regular edge-transitive graph of valency three or four has a semiregular automorphism

    Semiregular automorphisms of cubic vertex-transitive graphs

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    We characterise connected cubic graphs admitting a vertex- transitive group of automorphisms with an abelian normal subgroup that is not semiregular. We illustrate the utility of this result by using it to prove that the order of a semiregular subgroup of maximum order in a vertex-transitive group of automorphisms of a connected cubic graph grows with the order of the graph.Comment: This paper settles Problem 6.3 in P. Cameron, M. Giudici, G. Jones, W. Kantor, M. Klin, D. Maru\v{s}i\v{c}, L. A. Nowitz, Transitive permutation groups without semiregular subgroups, J. London Math. Soc. 66 (2002), 325--33

    Cubic vertex-transitive non-Cayley graphs of order 12p

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    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 12p12p, where pp is a prime, is given. As a result, there are 1111 sporadic and one infinite family of such graphs, of which the sporadic ones occur when p=5p=5, 77 or 1717, and the infinite family exists if and only if p1 (mod4)p\equiv1\ (\mod 4), and in this family there is a unique graph for a given order.Comment: This paper has been accepted for publication in SCIENCE CHINA Mathematic

    Elusive groups of automorphisms of digraphs of small valency

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    A transitive permutation group is called elusive if it contains no semiregular element. We show that no group of automorphisms of a connected graph of valency at most four is elusive and determine all the elusive groups of automorphisms of connected digraphs of out-valency at most three.Comment: To appear in the European Journal of Combinatoric

    On basic graphs of symmetric graphs of valency five

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    A graph \G is {\em symmetric} or {\em arc-transitive} if its automorphism group \Aut(\G) is transitive on the arc set of the graph, and \G is {\em basic} if \Aut(\G) has no non-trivial normal subgroup NN such that the quotient graph \G_N has the same valency with \G. In this paper, we classify symmetric basic graphs of order 2qpn2qp^n and valency 5, where q<pq<p are two primes and nn is a positive integer. It is shown that such a graph is isomorphic to a family of Cayley graphs on dihedral groups of order 2q2q with 5\di (q-1), the complete graph K6K_6 of order 66, the complete bipartite graph K5,5K_{5,5} of order 10, or one of the nine sporadic coset graphs associated with non-abelian simple groups. As an application, connected pentavalent symmetric graphs of order kpnkp^n for some small integers kk and nn are classified

    A classification of nilpotent 3-BCI groups

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    Given a finite group GG and a subset SG,S\subseteq G, the bi-Cayley graph \bcay(G,S) is the graph whose vertex set is G×{0,1}G \times \{0,1\} and edge set is {{(x,0),(sx,1)}:xG,sS}\{\{(x,0),(s x,1)\} : x \in G, s\in S \}. A bi-Cayley graph \bcay(G,S) is called a BCI-graph if for any bi-Cayley graph \bcay(G,T), \bcay(G,S) \cong \bcay(G,T) implies that T=gSαT = g S^\alpha for some gGg \in G and \alpha \in \aut(G). A group GG is called an mm-BCI-group if all bi-Cayley graphs of GG of valency at most mm are BCI-graphs.In this paper we prove that, a finite nilpotent group is a 3-BCI-group if and only if it is in the form U×V,U \times V, where UU is a homocyclic group of odd order, and VV is trivial or one of the groups Z2r,\Z_{2^r}, Z2r\Z_2^r and \Q_8

    Cubic arc-transitive kk-circulants

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    For an integer k1k\geq 1, a graph is called a kk-circulant if its automorphism group contains a cyclic semiregular subgroup with kk orbits on the vertices. We show that, if kk is even, there exist infinitely many cubic arc-transitive kk-circulants. We conjecture that, if kk is odd, then a cubic arc-transitive kk-circulant has order at most 6k26k^2. Our main result is a proof of this conjecture when kk is squarefree and coprime to 66

    Bipartite bi-Cayley graphs over metacyclic groups of odd prime-power order

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    A graph Γ\Gamma is a bi-Cayley graph over a group GG if GG is a semiregular group of automorphisms of Γ\Gamma having two orbits. Let GG be a non-abelian metacyclic pp-group for an odd prime pp, and let Γ\Gamma be a connected bipartite bi-Cayley graph over the group GG. In this paper, we prove that GG is normal in the full automorphism group Aut(Γ){\rm Aut}(\Gamma) of Γ\Gamma when GG is a Sylow pp-subgroup of Aut(Γ){\rm Aut}(\Gamma). As an application, we classify half-arc-transitive bipartite bi-Cayley graphs over the group GG of valency less than 2p2p. Furthermore, it is shown that there are no semisymmetric and no arc-transitive bipartite bi-Cayley graphs over the group GG of valency less than pp.Comment: 20 pages, 1 figur

    On edge-primitive and 2-arc-transitive graphs

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    A graph is edge-primitive if its automorphism group acts primitively on the edge set. In this short paper, we prove that a finite 2-arc-transitive edge-primitive graph has almost simple automorphism group if it is neither a cycle nor a complete bipartite graph. We also present two examples of such graphs, which are 3-arc-transitive and have faithful vertex-stabilizers
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