Higher chromatic numbers χs of simplicial complexes naturally
generalize the chromatic number χ1 of a graph. In any fixed dimension
d, the s-chromatic number χs of d-complexes can become arbitrarily
large for s≤⌈d/2⌉ [6,18]. In contrast, χd+1=1, and only
little is known on χs for ⌈d/2⌉<s≤d.
A particular class of d-complexes are triangulations of d-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 χ2 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 χ2 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 f=(127,8001,5334)
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 S3 with face vector
f=(167,1579,2824,1412) and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio