Related papers: On the CNF-complexity of bipartite graphs containi…

We prove that there exist infinite families of regular bipartite Ramanujan graphs of every degree bigger than 2. We do this by proving a variant of a conjecture of Bilu and Linial about the existence of good 2-lifts of every graph. We also…

Mixed graphs can be seen as digraphs with arcs and edges (or digons, that is, two opposite arcs). In this paper, we consider the case where such graphs are bipartite and in which the undirected and directed degrees are one. The best graphs,…

For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erd\H{o}s and Simonovits \cite{ESi1} conjectured that for…

Let $k$ and $l$ be integers, both at least 2. A $(k,l)$-bipartite graph is an $l$-regular bipartite multigraph with coloured bipartite sets of size $k$. Define $\chi(k,l)$ and $\mu(k,l)$ to be the minimum and maximum order of automorphism…

Let $G$ be a dense graph with good expansion properties and not too close to being bipartite. Let $\boldsymbol d$ be a graphical degree sequence. Under very weak conditions, we find the number of subgraphs of $G$ with degree sequence…

Topological drawings are natural representations of graphs in the plane, where vertices are represented by points, and edges by curves connecting the points. Topological drawings of complete graphs and of complete bipartite graphs have been…

We prove that any triangle-free graph on $n$ vertices with minimum degree at least $d$ contains a bipartite induced subgraph of minimum degree at least $d^2/(2n)$. This is sharp up to a logarithmic factor in $n$. Relatedly, we show that the…

Let $G$ be a connected graph. If $G$ contains a matching of size $k$, and every matching of size $k$ is contained in a perfect matching of $G$, then $G$ is said to be \emph{$k$-extendable}. A $k$-regular spanning subgraph of $G$ is called a…

An $(a,b)$-biregular bipartite graph is a bipartite graph with bipartition $(X, Y)$ such that each vertex in $X$ has degree $a$ and each vertex in $Y$ has degree $b$. By the bipartite expander mixing lemma, biregular bipartite graphs have…

We study the problem of Minimum $k$-Critical Bipartite Graph of order $(n,m)$ - M$k$CBG-$(n,m)$: to find a bipartite $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical bipartite, and the tuple $(|E|, \Delta_U,…

A bipartite graph G is known to be Pfaffian if and only if it does not contain an even subdivision H of $K_{3,3}$ such that $G - VH$ contains a 1-factor. However a general characterisation of Pfaffian graphs in terms of forbidden subgraphs…

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 study the problem of finding a minimum $k$-critical-bipartite graph of order $(n,m)$: a bipartite graph $G=(U,V;E)$, with $|U|=n$, $|V|=m$, and $n>m>1$, which is $k$-critical-bipartite, and the tuple $(|E|, \Delta_U, \Delta_V)$, where…

We consider the algorithmic problem of finding large \textit{balanced} independent sets in sparse random bipartite graphs, and more generally the problem of finding independent sets with specified proportions of vertices on each side of the…

A graph $G$ is called $C_{2k+1}$-free if it does not contain any cycle of length $2k+1$. In 1981, Haggkvist, Faudree and Schelp showed that every $n$-vertex triangle-free graph with more than $\frac{(n-1)^2}{4}+1$ edges is bipartite. In…

For the set of graphs with a given degree sequence, consisting of any number of $2's$ and $1's$, and its subset of bipartite graphs, we characterize the optimal graphs who maximize and minimize the number of $m$-matchings. We find the…

For a positive integer $k\ge 1$, a graph $G$ is $k$-stepwise irregular ($k$-SI graph) if the degrees of every pair of adjacent vertices differ by exactly $k$. Such graphs are necessarily bipartite. Using graph products it is demonstrated…

The paper consider an equivalence relation in the set of vertices of a bipartite graph. Some numerical characteristics showing the cardinality of equivalence classes are introduced. A combinatorial identity that is in relationship to these…

We consider the bipartite version of the {\it degree/diameter problem}, namely, given natural numbers $d\ge2$ and $D\ge2$, find the maximum number $\N^b(d,D)$ of vertices in a bipartite graph of maximum degree $d$ and diameter $D$. In this…

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, \Delta)$ be the largest $k$…