In this paper we give an explicit formula for the interlace polynomial at x = −1 for any graph, and as a result prove a conjecture of Arratia et al. that states that it is always of the form ±2 s . We also give a description of the graphs for which s is maximal.
We consider generalizations of the Tutte polynomial on multigraphs obtained by keeping the main recurrence relation T(G)=T(GÂe)+T(G&e) for e # E(G) neither a bridge nor a loop and dropping the relations for bridges and loops. Our first aim is to find the universal invariant satisfying these conditions, from which all others may be obtained. Surprisingly, this turns out to be the universal V-function Z of Tutte (1947, Proc. Cambridge Philos. Soc. 43, 26 40) defined to obey the same relation for bridges as well. We also obtain a corresponding result for graphs with colours on the edges and describe the universal coloured V-function, which is more complicated than Z.Extending results of Tutte (1974, J. Combin. Theory Ser. B 16, 168 174) and Brylawski (1981, J. Combin. Theory Ser. B 30, 233 246), we give a simple proof that there are non-isomorphic graphs of arbitrarily high connectivity with the same Tutte polynomial and the same value of Z. We conjecture that almost all graphs are determined by their chromatic or Tutte polynomials and provide mild evidence to support this.
Academic Press
In Croot, Lev and Pach's groundbreaking work [2], the authors showed that a subset of Z n 4 without an arithmetic progression of length 4 must be of size at most 3.1 n . No prior upper bound of the form (|G| − ε) n was known for the corresponding question in G n for any abelian group G containing elements of order greater than two.Refining the technique in [2], Ellenberg and Gijswijt [3] showed that a subset of Z n 3 c 2018 Luke Pebody c b Licensed under a Creative Commons Attribution License (CC-BY)
Given a subset S of an abelian group G and an integer k 1, the k-deck of S is the function that assigns to every T ⊆ G with at most k elements the number of elements g ∈ G with g + T ⊆ S. The reconstruction problem for an abelian group G asks for the minimal value of k such that every subset S of G is determined, up to translation, by its k-deck. This minimal value is the set-reconstruction number r set (G) of G; the corresponding value for multisets is the reconstruction number r(G). Previous work had given bounds for the set-reconstruction number of cyclic groups: Alon, Caro, Krasikov and Roditty [1] showed that r set (Z n) < log 2 n and Radcliffe and Scott [15] that r set (Z n) < 9 ln n ln ln n. We give a precise evaluation of r(G) for all abelian groups G and deduce that r set (Z n) 6.
Abstract. We prove that every finite subset of the plane is reconstructible from the multiset of its subsets of at most 18 points, each given up to rigid motion. We also give some results concerning the reconstructibility of infinite subsets of the plane.
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