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A generalization of the chromatic number of a graph is introduced such that the colors are integers modulo n, and the colors on adjacent vertices are required to be as far apart as possible.
Several problems concerning the distribution of cycle lengths in a graph have been proposed by P. Erdös and colleagues. In this note two variations of the following such question are answered. In a simple graph where every vertex has degree at least three, must there exist two cycles whose lengths differ by one or two? c © 1998 John Wiley & Sons, Inc. J(More)
This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents(More)
  • A. Vince
  • Discrete Applied Mathematics
  • 2004
Integrity, a measure of network reliability, is de-ned as I(G) = min S⊂V {|S|+ m(G − S)}; where G is a graph with vertex set V and m(G− S) denotes the order of the largest component of G − S. We prove an upper bound of the following form on the integrity of any cubic graph with n vertices: I(G)¡ 3 n+ O (√ n ) : Moreover, there exist an in-nite family of(More)
New integer invariants of a graph G, introduced by U. Oberst, are obtained as the elementary divisors of the Laplacian matrix of G. The theory of elementary divisors is developed in the context of regular matroids. It is shown that the elementary divisors of a graph are actually invariants of its underlying matroid. Regular matroids, in tum, are related to(More)
The problem of finding an approximation to a geometric line by a discrete line using pixels is ubiquitous in computer graphics applications. We show that this discrete line problem in R n+1, for grids of any shape, is equivalent to a geometry problem in Rn concerning the minimization of the distance that a certain type of closed polygonal path wanders from(More)
A rep-tiling ~is a self replicating, lattice tiling of R". Lattice tiling means a tiling by translates of a single compact tile by the points of a lattice, and self-replicating means that there is a non-singular linear map ¢: R"-o R" such that, for each T e J, the image 4~(T) is, in turn, tiled by Y-. This topic has recently come under investigation, not(More)
Although the representation of the real numbers in terms of a base and a set of digits has a long history, new questions arise even in the binary case – digits 0 and 1. A binary positional number system (binary radix system) with base equal to the golden ratio (1 + √ 5 )/2 is fairly well known. The main result of this paper is a construction of infinitely(More)