Vertex-primitive groups and graphs of order twice the product of two distinct odd primes

A non-Cayley number is an integer n for which there exists a vertex-transitive graph on n vertices which is not a Cayley graph. In this paper, we complete the determination of the non-Cayley numbers of the form 2pq, where p, q are distinct odd primes. Earlier work of Miller and the second author had dealt with all such numbers corresponding to vertex-transitive graphs admitting an imprimitive subgroup of automorphisms. This paper deals with the primitive case. First the primitive permutation groups of degree 2pq are classified. This depends on the finite simple group classification. Then each of these groups G is examined to determine whether there are any non-Cayley graphs which admit G as a vertex-primitive subgroup of automorphisms, and admit no imprimitive subgroups. The outcome is that 2pq is a non-Cayley number, where

Minimal generators of toric ideals of graphs

Let $I_G$ be the toric ideal of a graph $G$. We characterize in graph theoretical terms the primitive, the minimal, the indispensable and the fundamental binomials of the toric ideal $I_G$

On the diameter of the Kronecker product graph

Let $G_1$ and $G_2$ be two undirected nontrivial graphs. The Kronecker product of $G_1$ and $G_2$ denoted by $G_1\otimes G_2$ with vertex set $V(G_1)\times V(G_2)$, two vertices $x_1x_2$ and $y_1y_2$ are adjacent if and only if $(x_1,y_1)\in E(G_1)$ and $(x_2,y_2)\in E(G_2)$. This paper presents a formula for computing the diameter of $G_1\otimes G_2$ by means of the diameters and primitive exponents of factor graphs.Comment: 9 pages, 18 reference

Toric ideals associated with gap-free graphs

In this article we prove that every toric ideal associated with a gap-free graph $G$ has a squarefree lexicographic initial ideal. Moreover, in the particular case when the complementary graph of $G$ is chordal (i.e. when the edge ideal of $G$ has a linear resolution), we show that there exists a reduced Gr\"obner basis $\mathcal{G}$ of the toric ideal of $G$ such that all the monomials in the support of $\mathcal{G}$ are squarefree. Finally, we show (using work by Herzog and Hibi) that if $I$ is a monomial ideal generated in degree 2, then $I$ has a linear resolution if and only if all powers of $I$ have linear quotients, thus extending a result by Herzog, Hibi and Zheng.Comment: 13 pages, v2. To appear in Journal of Pure and Applied Algebr