2,742 research outputs found
On the chromatic number of cube-like graphs
AbstractA cube-like graph is a graph whose vertices are all 2n subsets of a set E of cardinality n, in which two vertices are adjacent if their symmetric difference is a member of a given specified collection of subsets of E. Many authors were interested in the chromatic number of such graphs and thought it was always a power of 2. Although this conjecture is false (we show a cube-like graph of chromatic number 7), we prove that there is no cube-like graph with chromatic number 3
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
Homomorphisms of binary Cayley graphs
A binary Cayley graph is a Cayley graph based on a binary group. In 1982,
Payan proved that any non-bipartite binary Cayley graph must contain a
generalized Mycielski graph of an odd-cycle, implying that such a graph cannot
have chromatic number 3. We strengthen this result first by proving that any
non-bipartite binary Cayley graph must contain a projective cube as a subgraph.
We further conjecture that any homo- morphism of a non-bipartite binary Cayley
graph to a projective cube must be surjective and we prove some special case of
this conjecture
On embeddings of CAT(0) cube complexes into products of trees
We prove that the contact graph of a 2-dimensional CAT(0) cube complex of maximum degree can be coloured with at most
colours, for a fixed constant . This implies
that (and the associated median graph) isometrically embeds in the
Cartesian product of at most trees, and that the event
structure whose domain is admits a nice labeling with
labels. On the other hand, we present an example of a
5-dimensional CAT(0) cube complex with uniformly bounded degrees of 0-cubes
which cannot be embedded into a Cartesian product of a finite number of trees.
This answers in the negative a question raised independently by F. Haglund, G.
Niblo, M. Sageev, and the first author of this paper.Comment: Some small corrections; main change is a correction of the
computation of the bounds in Theorem 1. Some figures repaire
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