Let Γ = (V,E) be a point-transitive reflexive relation. Let v ∈ V and put r = |Γ(v)|. Also assume Γ(v) ∩ Γ−(v) = {v}. Then |Γ(v) \ Γ(v)| ≥ r − 1. In particular we have |Γ(v)| ≥ 1 + (r − 1)j. The last result confirms a recent conjecture of Seymour in the case vertex-transitive graphs. Also it gives a short proof for the validity of the Caccetta-Häggkvist conjecture for vertex-transitive graphs a...

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V |) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that re...

We consider the problem of computing identifying codes of graphs and its fractional relaxation. The ratio between the size of optimal integer and fractional solutions is between 1 and 2 ln(|V |) + 1 where V is the set of vertices of the graph. We focus on vertex-transitive graphs for which we can compute the exact fractional solution. There are known examples of vertex-transitive graphs that re...

It is well-known that concentrators are sparse graphs of high connectivity, which play a key role in the construction of switching networks; and any semi-vertex transitive graph is isomorphic to a bi-coset graph. In this paper, we prove that random bi-coset graphs are almost always concentrators, and construct some examples of semi-vertex transitive concentrators.

We give examples of vertex-transitive graphs with non-monotonic chromatic difference sequences, disproving a conjecture of Albertson and Collins on the monotonicity of the chromatic difference sequence of vertex-transitive graphs, and answering a question of Zhou on the achievability of circulants.