369 research outputs found

    Chromatic Numbers of Simplicial Manifolds

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    Higher chromatic numbers χs\chi_s of simplicial complexes naturally generalize the chromatic number χ1\chi_1 of a graph. In any fixed dimension dd, the ss-chromatic number χs\chi_s of dd-complexes can become arbitrarily large for s≤⌈d/2⌉s\leq\lceil d/2\rceil [6,18]. In contrast, χd+1=1\chi_{d+1}=1, and only little is known on χs\chi_s for ⌈d/2⌉<s≤d\lceil d/2\rceil<s\leq d. A particular class of dd-complexes are triangulations of dd-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\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 χ2\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 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)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 S3S^3 with face vector f=(167,1579,2824,1412)f=(167,1579,2824,1412) and 2-chromatic number 5.Comment: 22 pages, 11 figures, revised presentatio

    Realizing the chromatic numbers and orders of spinal quadrangulations of surfaces

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    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

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    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

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    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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