369 research outputs found
Chromatic Numbers of Simplicial Manifolds
Higher chromatic numbers of simplicial complexes naturally
generalize the chromatic number of a graph. In any fixed dimension
, the -chromatic number of -complexes can become arbitrarily
large for [6,18]. In contrast, , and only
little is known on for .
A particular class of -complexes are triangulations of -manifolds. As a
consequence of the Map Color Theorem for surfaces [29], the 2-chromatic number
of any fixed surface is finite. However, by combining results from the
literature, we will see that for surfaces becomes arbitrarily large
with growing genus. The proof for this is via Steiner triple systems and is
non-constructive. In particular, up to now, no explicit triangulations of
surfaces with high were known.
We show that orientable surfaces of genus at least 20 and non-orientable
surfaces of genus at least 26 have a 2-chromatic number of at least 4. Via a
projective Steiner triple systems, we construct an explicit triangulation of a
non-orientable surface of genus 2542 and with face vector
that has 2-chromatic number 5 or 6. We also give orientable examples with
2-chromatic numbers 5 and 6.
For 3-dimensional manifolds, an iterated moment curve construction [18] along
with embedding results [6] can be used to produce triangulations with
arbitrarily large 2-chromatic number, but of tremendous size. Via a topological
version of the geometric construction of [18], we obtain a rather small
triangulation of the 3-dimensional sphere with face vector
and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio
Realizing the chromatic numbers and orders of spinal quadrangulations of surfaces
A method is suggested for construction of quadrangulations of the closed
orientable surface with given genus g and either (1) with given chromatic
number or (2) with given order allowed by the genus g. In particular, N.
Hartsfield and G. Ringel's results [Minimal quadrangulations of orientable
surfaces, J. Combin. Theory, Series B 46 (1989) 84-95] are generalized by way
of generating new minimal quadrangulations of infinitely many other genera.Comment: 6 pages. This version is only slightly different from the original
version submitted on 8 Jul 2012: the author's affiliation has been changed
and the presentation has been slightly improve
Colouring quadrangulations of projective spaces
A graph embedded in a surface with all faces of size 4 is known as a
quadrangulation. We extend the definition of quadrangulation to higher
dimensions, and prove that any graph G which embeds as a quadrangulation in the
real projective space P^n has chromatic number n+2 or higher, unless G is
bipartite. For n=2 this was proved by Youngs [J. Graph Theory 21 (1996),
219-227]. The family of quadrangulations of projective spaces includes all
complete graphs, all Mycielski graphs, and certain graphs homomorphic to
Schrijver graphs. As a corollary, we obtain a new proof of the Lovasz-Kneser
theorem
A new Kempe invariant and the (non)-ergodicity of the Wang-Swendsen-Kotecky algorithm
We prove that for the class of three-colorable triangulations of a closed
oriented surface, the degree of a four-coloring modulo 12 is an invariant under
Kempe changes. We use this general result to prove that for all triangulations
T(3L,3M) of the torus with 3<= L <= M, there are at least two Kempe equivalence
classes. This result implies in particular that the Wang-Swendsen-Kotecky
algorithm for the zero-temperature 4-state Potts antiferromagnet on these
triangulations T(3L,3M) of the torus is not ergodic.Comment: 37 pages (LaTeX2e). Includes tex file and 3 additional style files.
The tex file includes 14 figures using pstricks.sty. Minor changes. Version
published in J. Phys.
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