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Chromatic Numbers of Simplicial Manifolds

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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 ⌉ (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, 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 (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 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 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 S3 with face vector f= (167 , 1579 , 2824 , 1412) and 2-chromatic number 5.

OriginalsprogEngelsk
TidsskriftBeitraege zur Algebra und Geometrie
Antal sider35
ISSN0138-4821
DOI
StatusE-pub ahead of print - 2020

ID: 233783171