1,658 research outputs found

    Schrijver graphs and projective quadrangulations

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    In a recent paper [J. Combin. Theory Ser. B}, 113 (2015), pp. 1-17], the authors have extended the concept of quadrangulation of a surface to higher dimension, and showed that every quadrangulation of the nn-dimensional projective space PnP^n is at least (n+2)(n+2)-chromatic, unless it is bipartite. They conjectured that for any integers k≥1k\geq 1 and n≥2k+1n\geq 2k+1, the Schrijver graph SG(n,k)SG(n,k) contains a spanning subgraph which is a quadrangulation of Pn−2kP^{n-2k}. The purpose of this paper is to prove the conjecture

    Cochains in 2-TQFT

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    John C.Baez reinterpreted 2-dimensional and 3-dimensional topological quantum field theories (abbreviated as 2-TQFT and 3-TQFT) in "A prehistory of n-categorical physics"[JC11]. Inspired by his idea, this paper utilizes cochains to prove some properties of 2-TQFT and 3-TQFT. We also prove that cochains form an A∞A_{\infty} algebra under certain conditions.Comment: 5 pages, 3 figure

    A Geometric Approach to Combinatorial Fixed-Point Theorems

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    We develop a geometric framework that unifies several different combinatorial fixed-point theorems related to Tucker's lemma and Sperner's lemma, showing them to be different geometric manifestations of the same topological phenomena. In doing so, we obtain (1) new Tucker-like and Sperner-like fixed-point theorems involving an exponential-sized label set; (2) a generalization of Fan's parity proof of Tucker's Lemma to a much broader class of label sets; and (3) direct proofs of several Sperner-like lemmas from Tucker's lemma via explicit geometric embeddings, without the need for topological fixed-point theorems. Our work naturally suggests several interesting open questions for future research.Comment: 10 pages; an extended abstract appeared at Eurocomb 201
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