Karen L. Collins

Homomorphisms of 3-chromatic graphs

Abstract This paper examines the effect of a graph homomorphism upon the chromatic difference sequence of a graph. Our principal result (Theorem 2) provides necessary conditions for the existence of a homomorphism onto a prescribed target. As a consequence we note that iterated cartesian products of the Petersen graph form an infinite family of vertex transitive graphs no one of which is the homomorphic image of any other. We also prove that there is a unique minimal element in the homomorphism order of 3-chromatic graphs with non-monotonic chromatic difference sequences (Theorem 1). We include a brief guide to some recent papers on graph homomorphisms.

Duality and perfection for edges in cliques

Abstract Given a graph G , let K ( G ) denote the graph whose vertices correspond with the edges of G . Two vertices of K ( G ) are joined by an edge if the corresponding edges in G are contained in a clique. This paper investigates some properties of G which force duality theorems for K ( G ).

The number of Hamiltonian paths in a rectangular grid

Abstract It is easy to find out which rectangular m vertex by n vertex grids have a Hamiltonian path from one corner to another using a checkerboard argument. However, it is quite difficult in general to count the total number of such paths. In this paper we give generating function answers for grids with fixed m for m = 1, 2, 3, 4, 5.

Circulants and Sequences

A graph G is stable if its normalized chromatic difference sequence is equal to the normalized chromatic difference sequence of G X G, the Cartesian product of G with itself. Let

Factoring distance matrix polynomials

\alpha

On a conjecture of Graham and Lova´sz about distance matrices

be the independence number of G and let

A lower bound for 0,1, * tournament codes

\omega

Planar lattices are lexicographically shellable

be its clique number. Suppose that G has n vertices. We show that the first

An inversion relation of multinomial type

\omega

Split graphs and Nordhaus-Gaddum graphs

terms of the normalized chromatic difference sequence of a stable graph G must be