85 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
Local chromatic number of quadrangulations of surfaces
The local chromatic number of a graph G, as introduced in [4], is the minimum integer k such that G admits a proper coloring (with an arbitrary number of colors) in which the neighborhood of each vertex uses less than k colors. In [17] a connection of the local chromatic number to topological properties of (a box complex of) the graph was established and in [18] it was shown that a topological condition implying the usual chromatic number being at least four has the stronger consequence that the local chromatic number is also at least four. As a consequence one obtains a generalization of the following theorem of Youngs [19]: If a quadrangulation of the projective plane is not bipartite it has chromatic number four. The generalization states that in this case the local chromatic number is also four.
Both papers [1] and [13] generalize Youngs’ result to arbitrary non-orientable surfaces
replacing the condition of the graph being not bipartite by a more technical condition of
an odd quadrangulation. This paper investigates when these general results are true for the
local chromatic number instead of the chromatic number. Surprisingly, we find out that
(unlike in the case of the chromatic number) this depends on the genus of the surface. For
the non-orientable surfaces of genus at most four, the local chromatic number of any odd
quadrangulation is at least four, but this is not true for non-orientable surfaces of genus 5
or higher.
We also prove that face subdivisions of odd quadrangulations and Fisk triangulations of
arbitrary surfaces exhibit the same behavior for the local chromatic number as they do for
the usual chromatic number
Note On 6-regular Graphs On The Klein Bottle
Altshuler classified six regular graphs on the torus, but Thomassen and Negami gave different classifications for six regular graphs on the Klein bottle. In this note, we unify those two classifications, pointing out their difference and similarity
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