Related papers: Bipartite graphs with a perfect matching and digra…
We discuss the question whether the existence of perfect matchings in a cubic graph can be seen from the spectrum of its adjacency matrix. For regular graphs in general and for three edge-disjoint perfect matchings in a cubic graph (that…
A graph G is equimatchable if every maximal matching of G has the same cardinality. In this paper, we investigate equimatchable graphs such that the removal of any edge harms the equimatchability, called edge-critical equimatchable graphs…
The set of all perfect matchings of a plane (weakly) elementary bipartite graph equipped with a partial order is a poset, moreover the poset is a finite distributive lattice and its Hasse diagram is isomorphic to $Z$-transformation directed…
A matching in a bipartite graph with parts X and Y is called envy-free if no unmatched vertex in X is a adjacent to a matched vertex in Y. Every perfect matching is envy-free, but envy-free matchings exist even when perfect matchings do…
It is not hard to find many complete bipartite graphs which are not determined by their spectra. We show that the graph obtained by deleting an edge from a complete bipartite graph is determined by its spectrum. We provide some graphs, each…
We investigate the terminal-pairibility problem in the case when the base graph is a complete bipartite graph, and the demand graph is also bipartite with the same color classes. We improve the lower bound on maximum value of $\Delta(D)$…
Bandelt and Mulder's structural characterization of Bipartite Distance Hereditary graphs asserts that such graphs can be built inductively starting from a single vertex and by repeatedly adding either pending vertices or twins (i.e.,…
Barnette's Conjecture claims that all cubic, 3-connected, planar, bipartite graphs are Hamiltonian. We give a translation of this conjecture into the matching-theoretic setting. This allows us to relax the requirement of planarity to give…
We enumerate factorisations of the complete bipartite graph into spanning semiregular graphs in several cases, including when the degrees of all the factors except one or two are small. The resulting asymptotic behaviour is seen to…
A graph $G$ is said to be $k$-extendable if every matching of size $k$ in $G$ can be extended to a perfect matching of $G$, where $k$ is a positive integer. We say $G$ is $1$-excludable if for every edge $e$ of $G$, there exists a perfect…
We develop a bipartite rigidity theory for bipartite graphs parallel to the classical rigidity theory for general graphs, and define for two positive integers $k,l$ the notions of $(k,l)$-rigid and $(k,l)$-stress free bipartite graphs. This…
In the final project paper we consider a graph parameter called readability. Motivation for readability comes from bioinformatics applications. Graphs arising in problems related to genome sequencing are of small readability, which…
In the present paper, exterior algebraic shifting and symmetric algebraic shifting of bipartite graphs and chordal graphs are studied. First, we will determine the symmetric algebraic shifted graph of complete bipartite graphs. It turns out…
A graph $G$ is called \emph{symmetric with respect to a functional $F_G(P)$} defined on the set of all the probability distributions on its vertex set if the distribution $P^*$ maximizing $F_G(P)$ is uniform on $V(G)$. Using the…
The bipartite graph is a ubiquitous data structure that can model the relationship between two entity types: for instance, users and items, queries and webpages. In this paper, we study the problem of ranking vertices of a bipartite graph,…
The Dulmage--Mendelsohn decomposition (or the DM-decomposition) gives a unique partition of the vertex set of a bipartite graph reflecting the structure of all the maximum matchings therein. A bipartite graph is said to be DM-irreducible if…
We study the graphs formed from instances of the stable matching problem by connecting pairs of elements with an edge when there exists a stable matching in which they are matched. Our results include the NP-completeness of recognizing…
We say that a digraph $D$ is competitive if any pair of vertices has a common out-neighbor in $D$ and that a graph $G$ is competitively orientable if there exists a competitive orientation of $G$. The notion of competitive digraphs arose…
Combinatorial properties of some ideals related to strong quasi-n-partites graphs are examined. We prove that the edge ideal of a strong quasi-n-partite graph is not integrally closed and we give an expression for its integral closure.…
We study the connection between mixing properties for bipartite graphs and materialization of the mutual information in one-shot settings. We show that mixing properties of a graph imply impossibility to extract the mutual information…