35 research outputs found

    Clique number of tournaments

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    We introduce the notion of clique number of a tournament and investigate its relation with the dichromatic number. In particular, it permits defining \dic-bounded classes of tournaments, which is the paper's main topic

    Dilworth rate: a generalization of Witsenhausen's zero-error rate for directed graphs

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    Chromatic roots are dense in the whole complex plane

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    I show that the zeros of the chromatic polynomials P_G(q) for the generalized theta graphs \Theta^{(s,p)} are, taken together, dense in the whole complex plane with the possible exception of the disc |q-1| < 1. The same holds for their dichromatic polynomials (alias Tutte polynomials, alias Potts-model partition functions) Z_G(q,v) outside the disc |q+v| < |v|. An immediate corollary is that the chromatic zeros of not-necessarily-planar graphs are dense in the whole complex plane. The main technical tool in the proof of these results is the Beraha-Kahane-Weiss theorem on the limit sets of zeros for certain sequences of analytic functions, for which I give a new and simpler proof.Comment: LaTeX2e, 53 pages. Version 2 includes a new Appendix B. Version 3 adds a new Theorem 1.4 and a new Section 5, and makes several small improvements. To appear in Combinatorics, Probability & Computin

    Dilworth Rate: A Generalization of Witsenhausen’s Zero-Error Rate for Directed Graphs

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    Master index of volumes 61–70

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