SIAM Journal on Discrete Mathematics
Journal of Algebraic Combinatorics: An International Journal
A new approach to the orientation of random hypergraphs
Proceedings of the twenty-third annual ACM-SIAM symposium on Discrete Algorithms
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The Heilmann-Lieb Theorem on (univariate) matching polynomials states that the polynomial @?"km"k(G)y^k has only real nonpositive zeros, in which m"k(G) is the number of k-edge matchings of a graph G. There is a stronger multivariate version of this theorem. We provide a general method by which ''theorems of Heilmann-Lieb type'' can be proved for a wide variety of polynomials attached to the graph G. These polynomials are multivariate generating functions for spanning subgraphs of G with certain weights and constraints imposed, and the theorems specify regions in which these polynomials are nonvanishing. Such theorems have consequences for the absence of phase transitions in certain probabilistic models for spanning subgraphs of G.