The Alon-Tarsi conjecture states that for even n, the number of even latin squares of order n differs from the number of odd latin squares of order n. Zappa [6] found a generalization of this conjecture which makes sense for odd orders. In this note we prove this extended Alon-Tarsi conjecture for prime orders p. By results of Drisko [2] and Zappa [6], this implies that both conjectures are tru...

We give numerical and theoretical evidence in support of the conjecture of Dressler that between any two positive integers having the same prime factors there is a prime. In particular, it is shown that the abc conjecture implies that the gap between two consecutive such numbers a < c is a1/2− , and it is shown that this lower bound is best possible. Dressler’s conjecture is verified for values...

A well-established research area in finite group theory consists in exploring the interplay between the structure of a group G and certain sets of positive integers, which are naturally associated to G. One of those sets, denoted by cs(G), is the set of conjugacy class sizes of G. In order to have a better understanding of the arithmetical structure of cs(G), it is useful to introduce two kind ...

Abstract. The m, n Turk’s Head Knot, THK(m, n), is an “alternating (m, n) torus knot.” We prove the Harary-Kauffman conjecture for all THK(m, n) except for the case where m ≥ 5 is odd and n ≥ 3 is relatively prime to m. We also give evidence in support of the conjecture in that case. Our proof rests on the observation that none of these knots have prime determinant except for THK(m, 2) when Pm ...

At the 1912 Fifth International Congress of Mathematicians (Cambridge), Landau mentioned four unattackable problems: Goldbach Conjecture; Twin Prime Conjecture; for each n, there is a prime p such that n2 < p < (n+1)2; there are infinitely many primes p of the form p = n2+1. As Erdős [6, preface] maintained five years ago, we still do not have a satisfactory solution for each of them. For large...

A graph is said to be symmetric if its automorphism group is transitive on its arcs. A complete classification is given of pentavalent symmetric graphs of order 24p for each prime p. It is shown that a connected pentavalent symmetric graph of order 24p exists if and only if p=2, 3, 5, 11 or 17, and up to isomorphism, there are only eleven such graphs.

A graph is called symmetric if its full automorphism group acts transitively on the set of arcs. The Cayley graph $Gamma=Cay(G,S)$ on group $G$ is said to be normal symmetric if $N_A(R(G))=R(G)rtimes Aut(G,S)$ acts transitively on the set of arcs of $Gamma$. In this paper, we classify all connected tetravalent normal symmetric Cayley graphs of order $p^2q$ where $p>q$ are prime numbers.

In this article we define a minor relation, which is stronger than the classical one, but too strong to become a well-quasi-order on the class of finite graphs. Nevertheless, with this terminology we are able to introduce a conjecture, which would imply the Lovasz conjecture and give an interesting insight on the symmetry of vertex-transitive graphs, if true. Though it could become an approach ...

The Hadwiger number η(G) of a graph G is defined as the largest integer n for which the complete graph on n nodes Kn is a minor of G. Hadwiger conjectured that for any graph G, η(G) ≥ χ(G),where χ(G) is the chromatic number of G. In this paper, we investigate the Hadwiger number with respect to the cartesian product operation on Graphs. As the main result of this paper, we show that for any two...

Ramanujan graphs, introduced by Lubotzky, Phillips and Sarnak, allow the design of efficient communication networks. In joint work with B. Jordan we gave a higher-dimensional generalization. Here we explain how one could use this generalization to construct efficient communication networks which allow for a number of verification protocols and for the distribution of information along several c...