In this paper, we prove that every vertex-transitive graph can be expressed as the edge-disjoint union of symmetric graphs. We define a multicycle graph and conjecture that every vertex-transitive graph cam be expressed as the edge-disjoint union of multicycles. We verify this conjecture for several subclasses of vertextransitive graphs, including Cayley graphs, multidimensional circulants, and...

Let Ω = {1, 2, . . . , n} where n 2. The shape of an ordered set partition P = (P1, . . . , Pk) of Ω is the integer partition λ = (λ1, . . . , λk) defined by λi = |Pi|. Let G be a group of permutations acting on Ω. For a fixed partition λ of n, we say that G is λ-transitive if G has only one orbit when acting on partitions P of shape λ. A corresponding definition can also be given when G is jus...

An automorphism of a graph is called quasi-semiregular if it fixes unique vertex the and its remaining cycles have same length. This kind symmetry graphs was first investigated by Kutnar, Malnič, Martínez Marušič in 2013, as generalization well-known problem regarding existence semiregular automorphisms vertex-transitive graphs. Symmetric valency three or four, admitting automorphism, been clas...

This paper classifies all finite edge colored graphs with doubly transitive automorphism groups. This result generalizes the classification of doubly transitive balanced incomplete block designs with 1 and doubly transitive one-factorizations of complete graphs. It also provides a classification of all doubly transitive symmetric association schemes. The classification of finite simple groups i...

We investigate subsets of the symmetric group with structure similar to that of a graph. The “trees” of these subsets then lead to minimal highly symmetric generating sets of the symmetric group. We show that there exist generating sets among these with edge-transitive Cayley graphs and investigate them in relation to the Lovasz conjecture.

This paper describes the construction of all the vertex-transitive graphs on 24 vertices, thus extending the currently available catalogues. This construction differs significantly from previous constructions of the vertex-transitive graphs of order up to 23 in that we are forced to use far more sophisticated group-theoretic techniques. We include an analysis of all the symmetric graphs on 24 v...

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...