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Let S={n1,n2,…,ns}, s≥2, be a set of positive integers. A mixed hypergraph H=(X,C,D) is a one-realization of S if for every i∈S, there exists a unique coloring (up to permutations of the colors) using precisely i colors and there are no other colorings.For any r≥3 and set of integers S, we show that any r-uniform one-realization of S is isomorphic to a subhypergraph of Hn1,…,ns2 which is a special r-uniform one-realization of S. [ABSTRACT FROM AUTHOR]

A graph is a brick if it is 3-connected and bicritical. A brick is minimal if for every edge, the deletion of it results in a graph that is not a brick. In this article, we prove that every minimal brick has at least four cubic vertices. [ABSTRACT FROM AUTHOR]

Let G be a (molecule) graph. A perfect matching, or Kekulé structure of G is a set of independent edges covering every vertex exactly once. Enumeration of Kekulé structures of a (molecule) graph is interest in chemistry, physics and mathematics. In this paper, we focus on some polyominos on the torus and obtain the explicit expressions on the number of the Kekulé structures of them. [ABSTRACT FROM AUTHOR]

Abstract: The problem of enumerating close-packed dimers, or perfect matchings, on a quadratic lattice embedded on the Klein bottle is considered. Thomassen [C. Thomassen, Tilings of the torus and the Klein Bottle and vertex-transitive graphs on a fixed surface, Trans. Amer. Math. Soc. 323(2) (1991) 605–635] characterized that there are six quadrilateral lattices embedded on the Klein bottle. Lu and Wu [W. T. Lu and F. Y. Wu, Close-packed dimers on nonorientable surfaces, Phys. Lett. A 293 (2002) 235–246] had obtained a expression for the number of close-packed dimers on one of them. In this paper we investigate four other embeddings and obtain explicit expressions of the numbers of close-packed dimers and free energy per dimer by enumerating Pfaffians. [Copyright &y& Elsevier]