12,247 research outputs found
On Packing Colorings of Distance Graphs
The {\em packing chromatic number} of a graph is the
least integer for which there exists a mapping from to
such that any two vertices of color are at distance at
least . This paper studies the packing chromatic number of infinite
distance graphs , i.e. graphs with the set of
integers as vertex set, with two distinct vertices being
adjacent if and only if . We present lower and upper bounds for
, showing that for finite , the packing
chromatic number is finite. Our main result concerns distance graphs with
for which we prove some upper bounds on their packing chromatic
numbers, the smaller ones being for :
if is odd and
if is even
A generalization of Zhu's theorem on six-valent integer distance graphs
Given a set of positive integers, the integer distance graph for has
the set of integers as its vertex set, where two vertices are adjacent if and
only if the absolute value of their difference lies in . In 2002, Zhu
completely determined the chromatic number of integer distance graphs when
has cardinality . Integer distance graphs can be defined equivalently as
Cayley graphs on the group of integers under addition. In a previous paper, the
authors develop general methods to approach the problem of finding chromatic
numbers of Cayley graphs on abelian groups. To each such graph one associates
an integer matrix. In some cases the chromatic number can be determined
directly from the matrix entries. In particular, the authors completely
determine the chromatic number whenever the matrix is of size --
precisely the size of the matrices associated to the graphs studied by Zhu. In
this paper, then, we demonstrate that Zhu's theorem can be recovered as a
special case of the authors' previous results.Comment: 6 page
Circular chromatic numbers of some distance graphs
AbstractGiven a set D of positive integers, the distance graph G(Z,D) has vertices all integers Z, and two vertices j and j′ in Z are adjacent if and only if |j-j′|∈D. This paper determines the circular chromatic numbers of some distance graphs
Chromatic numbers of Cayley graphs of abelian groups: A matrix method
In this paper, we take a modest first step towards a systematic study of
chromatic numbers of Cayley graphs on abelian groups. We lose little when we
consider these graphs only when they are connected and of finite degree. As in
the work of Heuberger and others, in such cases the graph can be represented by
an integer matrix, where we call the dimension and the
rank. Adding or subtracting rows produces a graph homomorphism to a graph with
a matrix of smaller dimension, thereby giving an upper bound on the chromatic
number of the original graph. In this article we develop the foundations of
this method. In a series of follow-up articles using this method, we completely
determine the chromatic number in cases with small dimension and rank; prove a
generalization of Zhu's theorem on the chromatic number of -valent integer
distance graphs; and provide an alternate proof of Payan's theorem that a
cube-like graph cannot have chromatic number 3.Comment: 17 page
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