Independent Sets in Tensor Products of Three Vertex-transitive Graphs

Independent sets in direct products of vertex-transitive graphs

Let G and H be two vertex-transitive graphs. It is easy to find that α(G × H) = |G||H | 2 if one of them is a bipartite graph. In this paper, we will identify the structure of the maximal-sized independent sets in G×H when one of them is a bipartite graph.

Independent sets of maximal size in tensor powers of vertex-transitive graphs

Let G be a connected, non-bipartite vertex-transitive graph. We prove that if the only independent sets of maximal cardinality in the tensor product G×G are the preimages of the independent sets of maximal cardinality in G under projections, then the same holds for all finite tensor powers of G, thus providing an affirmative answer to a question raised by Larose and Tardif [8].

Primitivity and independent sets in direct products of vertex-transitive graphs

Vertex-transitive CIS graphs

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...

Cores of Vertex Transitive Graphs

A core of a graph X is a vertex minimal subgraph to which X admits a homomorphism. Hahn and Tardif have shown that for vertex transitive graphs, the size of the core must divide the size of the graph. This motivates the following question: when can the vertex set of a vertex transitive graph be partitioned into sets each of which induce a copy of its core? We show that normal Cayley graphs and ...