Related papers: Bipartite Minors

In 2010, Mader [W. Mader, Connectivity keeping paths in $k$-connected graphs, J. Graph Theory 65 (2010) 61-69.] proved that every $k$-connected graph $G$ with minimum degree at least $\lfloor\frac{3k}{2}\rfloor+m-1$ contains a path $P$ of…

We show that if the two parts of a finite bipartite graph have the same degree sequence, then there is a bipartite graph, with the same degree sequences, which is symmetric, in that it has an involutive graph automorphism that interchanges…

We show that any $3$-connected cubic plane graph on $n$ vertices, with all faces of size at most $6$, can be made bipartite by deleting no more than $\sqrt{(p+3t)n/5}$ edges, where $p$ and $t$ are the numbers of pentagonal and triangular…

For a bipartite graph $H$, its linear threshold is the smallest real number $\sigma$ such that every bipartite graph $G = (U \sqcup V, E)$ with unbalanced parts $|V| \gtrsim |U|^\sigma$ and without a copy of $H$ must have a linear number of…

Huynh et al. recently showed that a countable graph $G$ which contains every countable planar graph as a subgraph must contain arbitrarily large finite complete graphs as topological minors, and an infinite complete graph as a minor. We…

Let $G$ be a group and $L(G)$ be the set of all subgroups of $G$. We introduce a bipartite graph $\mathcal{B}(G)$ on $G$ whose vertex set is the union of two sets $G \times G$ and $L(G)$, and two vertices $(a, b) \in G \times G$ and $H \in…

Unmixed bipartite graphs have been characterized by Ravadra and Villarreal independently. Our aim in this paper is to characterize unmixed r-partite graphs under a certain condition, witch is a generalization of villarreal's theorem on…

We investigate size Ramsey numbers involving bipartite graphs. It is proved that, if each forbidden graph is fixed or grows with n (in a certain uniform manner), then the extremal function has a linear asymptotics. The corresponding slope…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

A graph $G = (V, E)$ is said to be word-representable if there exists a word $w$ over the alphabet $V$ such that, for any two distinct letters $x, y \in V$, the letters $x$ and $y$ alternate in $w$ if and only if $xy \in E$. A graph is…

A classic result in extremal graph theory, known as Mantel's theorem, states that every non-bipartite graph of order $n$ with size $m>\lfloor \frac{n^{2}}{4}\rfloor$ contains a triangle. Lin, Ning and Wu [Comb. Probab. Comput. 30 (2021)…

A bipartite graph $G=(V,E)$ with $V=V_1\cup V_2$ is biregular if all the vertices of a stable set $V_i$ have the same degree $r_i$ for $i=1,2$. In this paper, we give an improved new Moore bound for an infinite family of such graphs with…

For any field $\mathbb{F}$ and polynomials $f_{2},f_{3}\in\mathbb{F}[x,y]$, let $\Gamma_{\mathbb{F}}(f_{2},f_{3})$ denote the bipartite graph with vertex partition $P\cup L$, where $P$ and $L$ are two copies of $\mathbb{F}^{3}$, and…

Tutte showed that $4$-connected planar graphs are Hamiltonian, but it is well known that $3$-connected planar graphs need not be Hamiltonian. We show that $K_{2,5}$-minor-free $3$-connected planar graphs are Hamiltonian. This does not…

In a simple drawing of a graph every pair of edges intersect each other in at most one point, which is either a common endvertex or a proper crossing. For each positive integer $n$, Negami identified a drawing $B_n$ of the complete…

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…

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…

In this purely experimental work we try to represent the set of plane maps with 3 vertices and 3 faces as a bipartite ribbon graph. In particular, this construction allows one to estimate the genus of the initial set.

A bipartite covering of a (multi)graph $G$ is a collection of bipartite graphs, so that each edge of $G$ belongs to at least one of them. The capacity of the covering is the sum of the numbers of vertices of these bipartite graphs. In this…

We first introduce the class of bipartite absolute retracts with respect to tree obstructions with at most $k$ leaves. Then, using the theory of homomorphism duality, we show that this class of absolute retracts coincides exactly with the…