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 CIS graph is a graph in which every maximal stable set and every maximal clique intersect. A graph is well-covered if all its maximal stable sets are of the same size, co-well-covered if its complement is well-covered, and vertex-transitive if, for every pair of vertices, there exists an automorphism of the graph mapping one to the other. We show that a vertex-transitive graph is CIS if and o...

In this paper, we will determine the full automorphism groups of rose window graphs that are not edge-transitive. As the full automorphism groups of edge-transitive rose window graphs have been determined, this will complete the problem of calculating the full automorphism group of rose window graphs. As a corollary, we determine which rose window graphs are vertex-transitive. Finally, we deter...

We prove bounds on the chromatic number χ of a vertex-transitive graph in terms of its clique number ω and maximum degree ∆. We conjecture that every vertex-transitive graph satisfies χ 6 max{ω, ⌈ 5∆+3 6 ⌉ }, and we prove results supporting this conjecture. Finally, for vertex-transitive graphs with ∆ > 13 we prove the Borodin–Kostochka conjecture, i.e., χ 6 max{ω,∆− 1}.

Graphs possessing a high degree of symmetry have often been considered in topological graph theory. For instance, a number of constructions of genus embeddings by means of current or voltage graphs is based on the observation that a graph can be represented as a Cayley graph for some group. Another kind of embedding problems where symmetrical graphs are encountered is connected with regular map...

The isomorphism problem of Cayley graphs has been well studied in the literature, such as characterizations of CI (DCI)-graphs and CI (DCI)-groups. In this paper, we generalize these to vertex-transitive graphs and establish parallel results. Some interesting vertex-transitive graphs are given, including a first example of connected symmetric non-Cayley non-GI-graph. Also, we initiate the study...

In a recent paper (arXiv:1505.01475 ) Estélyi and Pisanski raised a question whether there exist vertex-transitive Haar graphs that are not Cayley graphs. In this note we construct an infinite family of trivalent Haar graphs that are vertex-transitive but non-Cayley. The smallest example has 40 vertices and is the well-known Kronecker cover over the dodecahedron graph G(10, 2), occurring as the...

We introduce a construction called the fractional multiples of a graph. These graphs are used to settle a question raised by E. Welzl: We show that if G and H are vertex-transitive graphs such that there exists a homomorphism fromG toH but no homomorphism fromH to G, then there exists a vertex-transitive graph that is homomorphically “in between” G and H.

A subgroup G of automorphisms of a graph X is said to be 1 2-transitive if it is vertex and edge but not arc-transitive. The graph X is said to be 1 2-transitive if Aut X is 1 2-transitive. The graph X is called one-regular if Aut X acts regularly on the set arcs of X. The interplay of three diierent concepts of maps, one-regular graphs and 1 2-transitive group actions on graphs of valency 4 is...