Chromatic Numbers of Simplicial Manifolds

Higher chromatic numbers chi(s) of simplicial complexes naturally generalize the chromatic number.1 of a graph. In any fixed dimension d, the s-chromatic number chi(s) of d-complexes can become arbitrarily large for s <= [d/2] (Bing in The geometric topology of 3-manifolds, Colloquium Publications, vol 40, American Mathematical Society, Providence, 1983; Heise et al. in Discrete Comput Geom 52:663-679, 2014). In contrast, chi(d+1) = 1, and only little is known on chi(s) for [d/2] < s <= d. A particular class of d-complexes are triangulations of d-manifolds. As a consequence of theMap Color Theorem for surfaces (Ringel in Map color theorem, Grundlehren der mathematischen Wissenschaften, vol 209, Springer, Berlin, 1974), the 2-chromatic number of any fixed surface is finite. However, by combining results from the literature, we will see that chi(2) 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 chi(2) 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 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 (Heise et al. 2014) along with embedding results (Bing 1983) 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 Heise et al. (2014), we obtain a rather small triangulation of the 3-dimensional sphere S-3 with face vector f = (167, 1579, 2824, 1412) and 2-chromatic number 5.