On Finite Subnormal Cayley Graphs

In this paper we introduce and study a type of Cayley graph – subnormal Cayley graph. We prove that a subnormal 2-arc transitive Cayley graph is a normal Cayley graph or a normal cover of a complete bipartite graph $\mathbf{K}_{p^d,p^d}$ with $p$ prime. Then we obtain a generic method for constructing half-symmetric (namely edge transitive but not arc transitive) Cayley graphs.

On tetravalent s-regular Cayley graphs

A Cayley graph [Formula: see text] is said to be core-free if [Formula: see text] is core-free in some [Formula: see text] for [Formula: see text]. A graph [Formula: see text] is called [Formula: see text]-regular if [Formula: see text] acts regularly on its [Formula: see text]-arcs. It is shown in this paper that if [Formula: see text], then there exist no core-free tetravalent [Formula: see text]-regular Cayley graphs; and for [Formula: see text], every tetravalent [Formula: see text]-regular Cayley graph is a normal cover of one of the three known core-free graphs. In particular, a characterization of tetravalent [Formula: see text]-regular Cayley graphs is given.

Finite Edge-Transitive Oriented Graphs of Valency Four: a Global Approach

We develop a new framework for analysing finite connected, oriented graphs of valency four, which admit a vertex-transitive and edge-transitive group of automorphisms preserving the edge orientation. We identify a sub-family of `basic' graphs such that each graph of this type is a normal cover of at least one basic graph. The basic graphs either admit an edge-transitive group of automorphisms that is quasiprimitive or biquasiprimitive on vertices, or admit an (oriented or unoriented) cycle as a normal quotient. We anticipate that each of these additional properties will facilitate effective further analysis, and we demonstrate that this is so for the quasiprimitive basic graphs. Here we obtain strong restrictions on the group involved, and construct several infinite families of such graphs which, to our knowledge, are different from any recorded in the literature so far. Several open problems are posed in the paper.

Finite 1-Regular Cayley Graphs of Valency 5

Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency .

LOCALLY PRIMITIVE GRAPHS AND BIDIRECT PRODUCTS OF GRAPHS

We characterise regular bipartite locally primitive graphs of order 2pe, where p is prime. We show that either p=2 (this case is known by previous work), or the graph is a binormal Cayley graph or a normal cover of one of the basic locally primitive graphs; these are described in detail.

TETRAVALENT s-TRANSITIVE GRAPHS OF ORDER TWICE A PRIME POWER

A graph is s-transitive if its automorphism group acts transitively on s-arcs but not on (s+1)-arcs in the graph. Let X be a connected tetravalent s-transitive graph of order twice a prime power. In this paper it is shown that s=1,2,3 or 4. Furthermore, if s=2, then X is a normal cover of one of the following graphs: the 4-cube, the complete graph of order 5, the complete bipartite graph K5,5 minus a 1-factor, or K7,7 minus a point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,2); if s=3, then X is a normal cover of the complete bipartite graph of order 4; if s=4, then X is a normal cover of the point-hyperplane incidence graph of the three-dimensional projective geometry PG(2,3). As an application, we classify the tetravalent s-transitive graphs of order 2p2 for prime p.

LOCALLY PRIMITIVE GRAPHS OF PRIME-POWER ORDER

Let Γ be a finite connected undirected vertex transitive locally primitive graph of prime-power order. It is shown that either Γ is a normal Cayley graph of a 2-group, or Γ is a normal cover of a complete graph, a complete bipartite graph, or Σ×l, where Σ=Kpm with p prime or Σ is the Schläfli graph (of order 27). In particular, either Γ is a Cayley graph, or Γ is a normal cover of a complete bipartite graph.

Even Covering Properties and Somewhat Normal Spaces

A topological space X is said to be somewhat normal provided that for each open cover is a normal cover of X. We show that a completely regular somewhat normal space need not be normal, thereby answering a question of W. M. Fleischman. We note that a collectionwise normal somewhat normal space need not be almost 2-fully normal, as had previously been asserted, and that neither the perfect image nor the perfect preimage of a somewhat normal space has to be somewhat normal.