Abelian groups without 3-chromatic Cayley graphs · arXivDesk
2410.11028Oct 14, 202411 pages. This paper was formerly titled "Abelian groups with 3-chromatic Cayley graphs." Additionally, in the revised version, some typos and minor errors were corrected, and some references and a brief subsection on quadrangulations were added
Let G be an abelian group. The main theorem of this paper asserts that there exists a Cayley graph on G with chromatic number 3 if and only if G is not of exponent 1, 2, or 4
Nearby in the stack
. For connected Cayley graphs, we also show that this theorem holds when
G
is finitely generated. Although motivated by ideas from algebraic topology, our proof may be expressed purely combinatorially. As a by-product, we derive a topological result which is of independent interest. Suppose
X
is a connected non-bipartite graph, and let
N(X)
denote its neighborhood complex. We show that if the fundamental group
π1(N(X))
or first homology group
H1(N(X))
is torsion, then the chromatic number of
X
is at least
4
. This strengthens a special case of a classical result of Lovász, which derives the same conclusion if