In this paper we examine the orders of vertex-transitive self-complementary uniform hypergraphs. In particular, we prove that if there exists a vertex-transitive selfcomplementary k-uniform hypergraph of order n, where k = 2 or k = 2 + 1 and n ≡ 1 (mod 2), then the highest power of any prime dividing n must be congruent to 1 modulo 2. We show that this necessary condition is also sufficient in ...

A graph Γ is said to be locally (G, 2)-arc transitive for G a subgroup of Aut(Γ) if, for any vertex α of Γ, G is transitive on the 2-arcs of Γ starting at α. In this talk, we will discuss general results involving locally (G, 2)-arc transitive graphs and recent progress toward the classification of the locally (G, 2)-arc transitive graphs, where Sz(q) ≤ G ≤ Aut(Sz(q)), q = 2 for some k ∈ N. In ...

We show that an action of SL(2, p), p ≥ 7 an odd prime such that 4 6 | (p − 1), has exactly two orbital digraphs Γ1, Γ2, such that Aut(Γi) admits a complete block system B of p + 1 blocks of size 2, i = 1, 2, with the following properties: the action of Aut(Γi) on the blocks of B is nonsolvable, doubly-transitive, but not a symmetric group, and the subgroup of Aut(Γi) that fixes each block of B...

It is known that there are precisely three transitive permutation groups of degree 6 admit an invariant partition with parts size 2 such the kernel action on has order 4; these called A 4 ( ) , S d and c . For each L ∈ { } we construct infinite family finite connected 6-valent graphs Γ n N arc-transitive G ≤ Aut group induced by vertex-stabiliser v neighbourhood a vertex isomorphic to L, ∣ expo...

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-...

Let Γ = (V,E) be a point-transitive reflexive relation. Let v ∈ V and put r = |Γ(v)|. Also assume Γ(v) ∩ Γ−(v) = {v}. Then |Γ(v) \ Γ(v)| ≥ r − 1. In particular we have |Γ(v)| ≥ 1 + (r − 1)j. The last result confirms a recent conjecture of Seymour in the case vertex-transitive graphs. Also it gives a short proof for the validity of the Caccetta-Häggkvist conjecture for vertex-transitive graphs a...

A graph Γ is self-complementary if its complement is isomorphic to the graph itself. An isomorphism that maps Γ to its complement Γ is called a complementing isomorphism. The majority of this dissertation is intended to present my research results on the study of self-complementary vertex-transitive graphs. I will provide an introductory mini-course for the backgrounds, and then discuss four pr...

A graph G admits an H-covering if every edge of belongs to a subgraph isomorphic given H. is said be H-magic there exists bijection f:V(G)∪E(G)→{1,2,…,|V(G)|+|E(G)|} such that wf(H′)=∑v∈V(H′)f(v)+∑e∈E(H′)f(e) constant, for H′ In particular, H-supermagic f(V(G))={1,2,…,|V(G)|}. When H complete K2, H-(super)magic labeling edge-(super)magic labeling. Suppose F-covering and two graphs F We define (...

Two questions are considered, namely (i) How many colors are needed for a coloring of the n-cube without monochromatic quadrangles or hexagons? We show that four colors suffice and thereby settle a problem of Erdos. (ii) Which vertex-transitive induced subgraphs does a hypercube have? An interesting graph has come up in this context: If we delete a Hamming code from the 7-cube, the resulting gr...

J.M. Howie, the influential St Andrews semigroupist, claimed that we value an area of pure mathematics to the extent that (a) it gives rise to arguments that are deep and elegant, and (b) it has interesting interconnections with other parts of pure mathematics. This paper surveys some recent results on the transformation semigroup generated by a permutation group G and a single non-permutation ...