1,149 research outputs found
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 -geodesic transitivity
For a graph , a positive integer and a subgroup G\leq
\Aut(\Gamma), we prove that is transitive on the set of -arcs of
if and only if has girth at least and is
transitive on the set of -geodesics of its line graph. As applications,
we first prove that the only non-complete locally cyclic -geodesic
transitive graphs are the complete multipartite graph and the
icosahedron. Secondly we classify 2-geodesic transitive graphs of valency 4 and
girth 3, and determine which of them are geodesic transitive
- …