17 research outputs found
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
This Week's Finds in Mathematical Physics (1-50)
These are the first 50 issues of This Week's Finds of Mathematical Physics,
from January 19, 1993 to March 12, 1995. These issues focus on quantum gravity,
topological quantum field theory, knot theory, and applications of
-categories to these subjects. However, there are also digressions into Lie
algebras, elliptic curves, linear logic and other subjects. They were typeset
in 2020 by Tim Hosgood. If you see typos or other problems please report them.
(I already know the cover page looks weird).Comment: 242 page