The aim of this paper is to give a coherent account of the problem of constructing cubic graphs with large girth. There is a well-defined integer µ 0 (g), the smallest number of vertices for which a cubic graph with girth at least g exists, and furthermore, the minimum value µ 0 (g) is attained by a graph whose girth is exactly g. The values of µ 0 (g) when… (More)
A proof that a concept is learnable provided the Vapnik-Chervonenkis dimension is finite is given. The proof is more explicit than previous proofs and introduces two new parameters which allow bounds on the sample size obtained to be improved by a factor of approximately 4log 2 (e).
We present exact calculations of the zero-temperature partition function for the q-state Potts antiferromagnet (equivalently, the chromatic polynomial) for two families of arbitrarily long strip graphs of the square lattice with periodic boundary conditions in the transverse direction and (i) periodic (ii) twisted periodic boundary conditions in the… (More)
The critical group of a graph is an abelian group that arises in several contexts, and there are some similarities with the groups that are used in cryptography. We construct a family of graphs with critical groups that are cyclic, and discuss the associated computational problems using algorithms based on the theory of 'chip-firing'.
This paper is motivated by a problem that arises in the study of partition functions of antiferromagnetic Potts models, including as a special case the chromatic polynomial. It relies on a theorem of Beraha, Kahane and Weiss, which asserts that the zeros of certain sequences of polynomials approach the curves on which a matrix has two eigenvalues with equal… (More)