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...

We introduce two variations of the cops and robber game on graphs. These games yield invariants in Z+∪{∞} for any connected graph Γ, weak cop number wCop(Γ) strong sCop(Γ). satisfy that sCop(Γ)≤wCop(Γ). Any is finite or a tree has one. new are preserved under small local perturbations graph, specifically, both numbers quasi-isometric More generally, we prove if Δ quasi-retract Γ then wCop(Δ)≤wC...

Metacirculants are a basic and well-studied family of vertex-transitive graphs, and weak metacirculants are generalizations of them. A graph is called a weak metacirculant if it has a vertex-transitive metacyclic automorphism group. This paper is devoted to the study of weak metacirculants with odd prime power order. We first prove that a weak metacirculant of odd prime power order is a metacir...

The present paper investigates arc-transtive graphs in term of their stability, and shows, somewhat contrary to expectations, that the property of instability is not as rare as previously thought. Until quite recently, the only known example of a finite, arc-transitive vertex-determining unstable graph was the underlying graph of the dodecahedron. This paper illustrates some methods for constru...

We prove an analog of results by Erdős-Ko-Rado and GreenwellLovász by characterising the maximum stable sets in special vertex-transitive subgraphs of powers of complete graphs, and proving that these graphs admit a unique optimal vertex colouring, up to permutation of the coordinates.

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...