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Bhatt and Leighton proved that the crossing number of a network (graph) is closely related to the minimum layout area required for the implementation of a VLSI circuit for that network. With this important application in mind, it makes most sense to analyze the crossing numbers of graphs with...
Let G be a graph associated with the hexagonal system of a benzenoid hydrocarbon. Let λ 1 ⩽ ⋯ ⩽ λ n be the set of eigenvalues of the adjacency matrix of G and consider the momenta M k = ∑ i = 1 n λ i k for k even. We show the lower bound M q 2 ( M r M s ) - 1 / 2 ⩽ E π ( G ) for the...
A spatial representation, R ( G ) , of a graph G , is an embedded image of G in R 3 . A set of cycles in R ( G ) can be thought of as a set of simple closed curves in R 3 and thus they may be regarded as a link in R 3 . A recent area of research investigates the dependence (or independence) of...
This paper surveys recent work in matroid representation theory and discusses a number of open problems.
We determine the facets of the polyhedral cone associated to the Rees algebra of the ideal generated by the square-free monomials of a fixed degree in a polynomial ring. Then we compute the a -invariant of those Rees algebras with respect to a certain grading.
The process called the chip-firing game has been around for no more than 20 years, but it has rapidly become an important and interesting object of study in structural combinatorics. The reason for this is partly due to its relation with the Tutte polynomial and group theory, but also because of...
In this expository paper we collect some combinatorial problems in the additive theory that can be easily solved in ordered Abelian groups. We study how such results, obtained by simple combinatorial arguments, can be extended to other Abelian groups. In many cases, best results can be obtained...
This paper discusses reformulations of the problem of coloring plane maps with four colors. We include discussion of the Eliahou–Kryuchkov conjecture, the Penrose formula, the vector cross-product formulation and the reformulations in terms of formations and factorizations due to G. Spencer-Brown.
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