Related papers: A precise condition for independent transversals i…
For a graph $G$ and partition $\mathcal{U}$ of its vertex set, an independent transversal of $(G, \mathcal{U})$ is an independent set of $G$ that contains one vertex from each block of $\mathcal{U}$. Buys, Kang, and Ozeki studied when a…
We give a sharp refinement of a result of Alon, Ben-Shimon and Krivelevich. This gives a sufficient condition for a finite sequence of positive integers to be the vertex degree list of both parts of a bipartite graph. The condition depends…
An \emph{independent transversal} (IT) in a graph $G$ with a given vertex partition $P$ is an independent set of vertices of $G$ (i.e. it induces no edges), that consists of one vertex from each part (\emph{block}) of $P$. Over the years,…
Given integers $r>d\ge 0$ and an $r$-partite graph, an independent $(r-d)$-transversal or $(r-d)$-IT is an independent set of size $r-d$ that intersects each part in at most one vertex. We show that every $r$-partite graph with maximum…
A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of each stable set, $V_1$ and $V_2$, have the same degree, $r$ and $s$, respectively. This paper studies difference sets derived from both Abelian and…
We study the appearance of the giant component in random subgraphs of a given large finite graph G=(V,E) in which each edge is present independently with probability p. We show that if G is an expander with vertices of bounded degree, then…
Let s<t be two fixed positive integers. We study what are the minimum degree conditions for a bipartite graph G, with both color classes of size n=k(s+t), which ensure that G has a K_{s,t}-factor. Exact result for large n is given. Our…
In this paper, two open conjectures are disproved. One conjecture regards independent coverings of sparse partite graphs, whereas the other conjecture regards orthogonal colourings of tree graphs. A relation between independent coverings…
K\H onig's theorem says that the vertex cover number of every bipartite graph is at most its matching number (in fact they are equal since, trivially, the matching number is at most the vertex cover number). An equivalent formulation of K\H…
Motivated both by recently introduced forms of list colouring and by earlier work on independent transversals subject to a local sparsity condition, we use the semi-random method to prove the following result. For any function $\mu$…
A hypergraph is \textit{bipartite with bipartition $(A, B)$} if every edge has exactly one vertex in $A$, and a matching in such a hypergraph is \textit{$A$-perfect} if it saturates every vertex in $A$. We prove an upper bound on the number…
Let $G$ be a bipartite graph with bipartition $(X,Y)$. Inspired by a hypergraph problem, we seek an upper bound on the number of disjoint paths needed to cover all the vertices of $X$. We conjecture that a Hall-type sufficient condition…
The problem of determining the optimal minimum degree condition for a balanced bipartite graph on 2ms vertices to contain m vertex disjoint copies of K_{s,s} was solved by Zhao. Later Hladk\'y and Schacht, and Czygrinow and DeBiasio…
By implementing algorithmic versions of Sapozhenko's graph container methods, we give new algorithms for approximating the number of independent sets in bipartite graphs. Our first algorithm applies to $d$-regular, bipartite graphs…
We prove a sharp Ore-type criterion for hamiltonicity of balanced bipartite digraphs: A bipartite digraph D, with colour classes of cardinality N, is hamiltonian if, for every pair of vertices u and v from opposite colour classes of D such…
A bihole in a bipartite graph $G$ with partite sets $A$ and $B$ is an independent set $I$ in $G$ with $|I\cap A|=|I\cap B|$. We prove lower bounds on the largest order of biholes in balanced bipartite graphs subject to conditions involving…
We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical…
Let $G = (V,E)$ be a simple finite graph. The corresponding bunkbed graph $G^\pm$ consists of two copies $G^+ = (V^+,E^+),G^- = (V^-,E^-)$ of $G$ and additional edges connecting any two vertices $v_+ \in V_+,v_- \in V_-$ that are the copies…
Motivated by problems modeling the spread of infections in networks, in this paper we explore which bipartite graphs are most resilient to widespread infections under various parameter settings. Namely, we study bipartite networks with a…
A graph with vertex set V and edge set E is called a (d,c)-expander if the maximum degree of a vertex is d and, for every subset W of V that has cardinality at most |V|/2, the number of edges between vertices in W and vertices outside of W…