4,518 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

    Tremain equiangular tight frames

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    Equiangular tight frames provide optimal packings of lines through the origin. We combine Steiner triple systems with Hadamard matrices to produce a new infinite family of equiangular tight frames. This in turn leads to new constructions of strongly regular graphs and distance-regular antipodal covers of the complete graph.Comment: 11 page

    Completion and deficiency problems

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    Given a partial Steiner triple system (STS) of order nn, what is the order of the smallest complete STS it can be embedded into? The study of this question goes back more than 40 years. In this paper we answer it for relatively sparse STSs, showing that given a partial STS of order nn with at most r≤εn2r \le \varepsilon n^2 triples, it can always be embedded into a complete STS of order n+O(r)n+O(\sqrt{r}), which is asymptotically optimal. We also obtain similar results for completions of Latin squares and other designs. This suggests a new, natural class of questions, called deficiency problems. Given a global spanning property P\mathcal{P} and a graph GG, we define the deficiency of the graph GG with respect to the property P\mathcal{P} to be the smallest positive integer tt such that the join G∗KtG\ast K_t has property P\mathcal{P}. To illustrate this concept we consider deficiency versions of some well-studied properties, such as having a KkK_k-decomposition, Hamiltonicity, having a triangle-factor and having a perfect matching in hypergraphs. The main goal of this paper is to propose a systematic study of these problems; thus several future research directions are also given
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