For any class C of bipartite graphs, we define quasi-C to be the class of all graphs G such that every bipartition of G belongs to C. This definition is motivated by a generalisation of the switch Markov chain on perfect matchings from bipartite graphs to nonbipartite graphs. The monotone graphs, also known as bipartite permutation graphs and proper interval bigraphs, are such a class of bipart...

In this paper, it is proved that let G be a bipartite graph with bipartition (X, Y ) and with a perfect matching M , let G be an n-extendable graph, then G is minimally n-extendable if and only if, for any two vertices x ∈ X and y ∈ Y such that xy ∈ E(G), there are exactly n internally disjoint (x, y) M-alternating paths P1, P2, . . . , Pn such that Pi (1 i n) starts and ends with edges in E(G)...

An analysis is conducted of the multipartite entanglement for Gaussian states generated by the parametric down-conversion of a femtosecond frequency comb. Using a recently introduced method for constructing optimal entanglement criteria, a family of tests is formulated for mode decompositions that extends beyond the traditional bipartition analyses. A numerical optimization over this family is ...

A shared binary decision diagram (SBDD) represents a multiple-output function, where nodes are shared among outputs. A partitioned SBDD usually consists of two or more SBDDs that often share nodes. The separate SBDDs are optimized independently, often resulting in a reduction in the number of nodes over a single SBDD. We show a method for partitioning a single SBDD into two parts that reduces t...

We study the orbits of G = GL(V ) in the enhanced nilpotent cone V ×N , where N is the variety of nilpotent endomorphisms of V . These orbits are parametrized by bipartitions of n = dimV , and we prove that the closure ordering corresponds to a natural partial order on bipartitions. Moreover, we prove that the local intersection cohomology of the orbit closures is given by certain bipartition a...

Let be a simple and connected graph. A k-vertex separator [k-edge separator] is a subset of vertices [edges] whose deletion separates the vertex [edge] set of into two parts of equal cardinality, that are at distance greater than k in . Here we investigate the relation between the cardinality of these cutsets and the laplacian spectrum of . As a consequence of the study, we obtain the well-know...

We investigate a multicommodity network design problem where a discrete set of technologies with step-increasing cost and capacity functions should be installed on the edges. This problem is a fundamental network design problem having many important applications in contemporary telecommunication networks. We describe an exact constraint generation approach and we show that the conjunctive use o...

Let G be an (edge-)colored graph. A heterochromatic matching of G is a matching in which no two edges have the same color. For a vertex v, let d(v) be the color degree of v. We show that if d(v) ≥ k for every vertex v of G, then G has a heterochromatic matching of size ⌈ 5k−3 12 ⌉ . For a colored bipartite graph with bipartition (X,Y ), we prove that if it satisfies a Hall-like condition, then ...

Hall’s Theorem is a basic result in Combinatorics which states that the obvious necesssary condition for a finite family of sets to have a transversal is also sufficient. We present a sufficient (but not necessary) condition on the sizes of the sets in the family and the sizes of their intersections so that a transversal exists. Using this, we prove that in a bipartite graph G (bipartition {A,B...

Assume that m and n are positive even integers with n 4. The honeycomb rectangular torus HReT(m,n) is recognized as another attractive alternative to existing torus interconnection networks in parallel and distributed applications. It is known that any HReT(m,n) is a 3-regular bipartite graph. We prove that any HReT(m,n) − e is hamiltonian for any edge e ∈ E(HReT(m,n)). Moreover, any HReT(m,n)−...