A decomposition of a graph is a partition of the edge set. One can also look at partitions of the arc set but in this talk we restrict our attention to edges. If each part of the decomposition is a spanning subgraph then we call the decomposition a factorisation and the parts are called factors. Decompositions are especially interesting when the subgraphs induced by each part are pairwise isomo...

A module of an undirected graph is a set X of nodes such for each node x not in X , either every member of X is adjacent to x, or no member of X is adjacent to x. There is a canonical linear-space representation for the modules of a graph, called the modular decomposition. Closely related to modular decomposition is the transitive orientation problem, which is the problem of assigning a directi...

We call an undirected graph X half-transitive if the automorphism group Aut X of X acts transitively on the vertex set and edge set but not on the set of ordered pairs of adjacent vertices of X. In this paper we determine all half-transitive graphs of order p3 and degree 4, where p is an odd prime; namely, we prove that all such graphs are Cayley graphs on the non-Abelian group of order p3 and ...

A graph is half-arc-transitive if its automorphism group acts transitively on its vertex set, edge set, but not arc set. Let p and q be primes. It is known that no tetravalent half-arc-transitive graphs of order 2p2 exist and a tetravalent half-arctransitive graph of order 4p must be non-Cayley; such a non-Cayley graph exists if and only if p − 1 is divisible by 8 and it is unique for a given o...

Abstract We prove that, if $$\varGamma $$ Γ is a finite connected 3-valent vertex-transitive, or 4-valent vertex- and edge-transitive graph, then either part of well-understood family graphs, every non-identity automorphism fixes at most 1/3 the edges. This answers question proposed by Primož Potočnik third a...

Let $G$ be a connected graph with minimum degree $delta$ and edge-connectivity $lambda$. A graph ismaximally edge-connected if $lambda=delta$, and it is super-edge-connected if every minimum edge-cut istrivial; that is, if every minimum edge-cut consists of edges incident with a vertex of minimum degree.In this paper, we show that a connected graph or a connected triangle-free graph is maximall...

An edge cut of a connected graph is 4-restricted if it disconnects this graph with each component having order at least four. The size of minimum 4-restricted edge cuts of graph G is called its 4-restricted edge connectivity and is denoted by 4(G). Let 4(G)=min{ (F ) : F is a connected vertex-induced subgraph of order four of graph G}, where (F ) denotes the number of edges of graph G with exac...

A regular and edge transitive graph which is not vertex transitive is said to be semisymmetric Every semisymmetric graph is necessarily bipartite with the two parts having equal size and the automorphism group acting transitively on each of these two parts A semisymmet ric graph is called biprimitive if its automorphism group acts prim itively on each part In this paper a classi cation of bipri...

The existence of a connected 12-regular {K4,K2,2,2}-ultrahomogeneous graph G is established, (i.e. each isomorphism between two copies of K4 or K2,2,2 in G extends to an automorphism of G), with the 42 ordered lines of the Fano plane taken as vertices. This graph G can be expressed in a unique way both as the edge-disjoint union of 42 induced copies of K4 and as the edge-disjoint union of 21 in...

The action of a subgroup G of automorphisms of a graph X is said to be 2 -transitive if it is vertexand edgebut not arc-transitive. In this case the graph X is said to be (G, 2)-transitive. In particular, X is 1 2 -transitive if it is (Aut X, 1 2)-transitive. The 2 -transitive action of G on X induces an orientation of the edges of X which is preserved by G. Let X have valency 4. An even length...