Related papers: Complete bipartite graphs flexible in the plane

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 introduce a notion of bipartite minors and prove a bipartite analog of Wagner's theorem: a bipartite graph is planar if and only if it does not contain $K_{3,3}$ as a bipartite minor. Similarly, we provide a forbidden minor…

A connected graph is called a bi-block graph if each of its blocks is a complete bipartite graph. Let $\mathcal{B}(\mathbf{k}, \alpha)$ be the class of bi-block graph on $\mathbf{k}$ vertices with given independence number $\alpha$. It is…

A vertex set $D$ in a finite undirected graph $G$ is an {\em efficient dominating set} (\emph{e.d.s.}\ for short) of $G$ if every vertex of $G$ is dominated by exactly one vertex of $D$. The \emph{Efficient Domination} (ED) problem, which…

In this paper, we generalize the notions of perfect matchings, perfect 2-matchings to perfect k-matchings and give a necessary and sufficient condition for existence of perfect k-matchings. For bipartite graphs, we show that this k-matching…

The $d$-dimensional algebraic connectivity $a_d(G)$ of a graph $G=(V,E)$ is a quantitative measure of its $d$-dimensional rigidity, defined in terms of the eigenvalues of stiffness matrices associated with different embeddings of the graph…

An old problem raised independently by Jacobson and Sch\"onheim asks to determine the maximum $s$ for which every graph with $m$ edges contains a pair of edge-disjoint isomorphic subgraphs with $s$ edges. In this paper we determine this…

In [7], Higashitani, Kummer, and Micha{\l}ek pose a conjecture about the symmetric edge polytopes of complete multipartite graphs and confirm it for a number of families in the bipartite case. We confirm that conjecture for a number of new…

Given a multigraph $G$, the all-terminal reliability $R(G,p)$ is the probability that $G$ remains connected under percolation with parameter $p$. Fixing the number of vertices $n$ and edges $m$, we investigate which graphs maximize $R(G,p)$…

Given a family $\mathcal{F}$ of bipartite graphs, the {\it Zarankiewicz number} $z(m,n,\mathcal{F})$ is the maximum number of edges in an $m$ by $n$ bipartite graph $G$ that does not contain any member of $\mathcal{F}$ as a subgraph (such…

We prove that any $n$-vertex graph whose complement is triangle-free contains $n^2/12-o(n^2)$ edge-disjoint triangles. This is tight for the disjoint union of two cliques of order $n/2$. We also prove a corresponding stability theorem, that…

A graph is $k$-planar if it can be drawn in the plane such that no edge is crossed more than $k$ times. While for $k=1$, optimal $1$-planar graphs, i.e., those with $n$ vertices and exactly $4n-8$ edges, have been completely characterized,…

Let G be a graph in a 3-manifold M. We compress the pair (M,G) along admissible 2-spheres as long as possible. What we get is a root of (M,G). Our main result is that for any pair (M,G) the root exists and is unique. As a corollary we get…

In this paper, we study $(1,2)$-step competition graphs of bipartite tournaments. A bipartite tournament means an orientation of a complete bipartite graph. We show that the $(1,2)$-step competition graph of a bipartite tournament has at…

The focus of this paper is on discussion of a catalog of a class of (3, g) graphs for even girth g. A (k, g) graph is a graph with regular degree k and girth g. This catalog is compared with other known lists of (3, g) graphs such as the…

We prove that a complete multipartite graph $K$ with $n>1$ vertices and $m$ edges can be decomposed into edge-disjoint Hamilton paths if and only if $\frac m{n-1}$ is an integer and the maximum degree of $K$ is at most $\frac {2m}{n-1}$.

There are extensive studies focusing on the application scenario that all the bipartite cohesive subgraphs need to be discovered in a bipartite graph. However, we observe that, for some applications, one is interested in finding bipartite…

For a graph $G = (V, E)$, the $\gamma$-graph of $G$ is the graph whose vertex set is the collection of minimum dominating sets, or $\gamma$-sets of $G$, and two $\gamma$-sets are adjacent if they differ by a single vertex and the two…

A graph is Schur-positive if its chromatic symmetric function expands non-negatively in the Schur basis. We determine a full Schur-positivity classification for complete multipartite graphs by showing that a complete multipartite graph…

Lovasz and Plummer conjectured in the 1970's that cubic bridgeless graphs have exponentially many perfect matchings. This conjecture has been verified for bipartite graphs by Voorhoeve in 1979, and for planar graphs by Chudnovsky and…