A note on pentavalent s-transitive graphs

AbstractA graph, with a group G of its automorphisms, is said to be (G,s)-transitive if G is transitive on s-arcs but not on (s+1)-arcs of the graph. Let X be a connected (G,s)-transitive graph for some s≥1, and let Gv be the stabilizer of a vertex v∈V(X) in G. In this paper, we determine the structure of Gv when X has valency 5 and Gv is non-solvable. Together with the results of Zhou and Feng [J.-X. Zhou, Y.-Q. Feng, On symmetric graphs of valency five, Discrete Math. 310 (2010) 1725–1732], the structure of Gv is completely determined when X has valency 5. For valency 3 or 4, the structure of Gv is known

Colouring Cayley Graphs

We will discuss three ways to bound the chromatic number on a Cayley graph. 1. If the connection set contains information about a smaller graph, then these two graphs are related. Using this information, we will show that Cayley graphs cannot have chromatic number three. 2. We will prove a general statement that all vertex-transitive maximal triangle-free graphs on n vertices with valency greater than n/3 are 3-colourable. Since Cayley graphs are vertex-transitive, the bound of general graphs also applies to Cayley graphs. 3. Since Cayley graphs for abelian groups arise from vector spaces, we can view the connection set as a set of points in a projective geometry. We will give a characterization of all large complete caps, from which we derive that all maximal triangle-free cubelike graphs on 2n vertices and valency greater than 2n/4 are either bipartite or 4-colourable

Line graphs and 22-geodesic transitivity

For a graph $\Gamma$, a positive integer $s$ and a subgroup G\leq \Aut(\Gamma), we prove that $G$ is transitive on the set of $s$-arcs of $\Gamma$ if and only if $\Gamma$ has girth at least $2(s-1)$ and $G$ is transitive on the set of $(s-1)$-geodesics of its line graph. As applications, we first prove that the only non-complete locally cyclic $2$-geodesic transitive graphs are the complete multipartite graph $K_{3[2]}$ and the icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and girth 3, and determine which of them are geodesic transitive