Related papers: Improved bounds for the crossing numbers of K_m,n …

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…

A graph is diameter-$k$-critical if its diameter equals $k$ and the deletion of any edge increases its diameter. The Murty-Simon Conjecture states that for any diameter-2-critical graph $G$ of order $n$, $e(G) \leq \lfloor…

In this note we provide an improved upper bound on the biplanar crossing number of the 8-dimensional hypercube. The $k$-planar crossing number of a graph $cr_k(G)$ is the number of crossings required when every edge of $G$ must be drawn in…

The crossing number of a graph $G$, ${\mbox{cr}}(G)$, is the minimum number of crossings, the pair-crossing number, ${\mbox{pcr}}(G)$, is the minimum number of pairs of crossing edges over all drawings of $G$. In this note we show that…

Let $P$ be a set of points in general position in the plane. Join all pairs of points in $P$ with straight line segments. The number of segment-crossings in such a drawing, denoted by $\crg(P)$, is the \emph{rectilinear crossing number} of…

Consider a $3$-uniform hypergraph of order $n$ with clique number $k$ such that the intersection of all its $k$-cliques is empty. Szemer\'edi and Petruska proved $n\leq 8m^2+3m$, for fixed $m=n-k$, and they conjectured the sharp bound $n…

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…

The Kneser graph $K(n,k)$ is the graph whose vertices are the $k$-elements subsets of an $n$-element set, with two vertices adjacent if the sets are disjoint. The square $G^2$ of a graph $G$ is the graph defined on $V(G)$ such that two…

A {\it krausz $(k,m)$-partition} of a graph $G$ is the partition of $G$ into cliques, such that any vertex belongs to at most $k$ cliques and any two cliques have at most $m$ vertices in common. The {\it $m$-krausz} dimension $kdim_m(G)$ of…

Given positive integers m,n,s,t, let z(m,n,s,t) be the maximum number of ones in a (0,1) matrix of size m-by-n that does not contain an all ones submatrix of size s-by-t. We find a flexible upper bound on z(m,n,s,t) that implies the known…

Scheinerman and Wilf (1994) assert that `an important open problem in the study of graph embeddings is to determine the rectilinear crossing number of the complete graph K_n.' A rectilinear drawing of K_n is an arrangement of n vertices in…

Given a fixed positive integer $k$, the $k$-planar local crossing number of a graph $G$, denoted by $\text{LCR}_k(G)$, is the minimum positive integer $L$ such that $G$ can be decomposed into $k$ subgraphs, each of which can be drawn in a…

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…

The Hamming graph $H(d,n)$ is the Cartesian product of $d$ complete graphs on $n$ vertices. Let $m=d(n-1)$ be the degree and $V = n^d$ be the number of vertices of $H(d,n)$. Let $p_c^{(d)}$ be the critical point for bond percolation on…

In the multiway cut problem, we are given an undirected graph with non-negative edge weights and a collection of $k$ terminal nodes, and the goal is to partition the node set of the graph into $k$ non-empty parts each containing exactly one…

This paper introduces the concepts of the augmented Zarankiewicz number $z_A(m,n)$ and the limited augmented Zarankiewicz number $z_L(m,n)$, which are natural combinatorial extensions of the classical Zarankiewicz number. These numbers…

Let $D$ be a straight-line drawing of a graph. The rectilinear 2-colored crossing number of $D$ is the minimum number of crossings between edges of the same color, taken over all possible 2-colorings of the edges of $D$. First, we show…

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 integers $m$ and $n$, we study the problem of finding good lower bounds for the size of progression-free sets in $(\mathbb{Z}_{m}^{n},+)$. Let $r_{k}(\mathbb{Z}_{m}^{n})$ denote the maximal size of a subset of $\mathbb{Z}_{m}^{n}$…

The number $Z(n):=\lfloor n/2\rfloor\lfloor (n-1)/2\rfloor$ is the smallest number of crossings in a simple planar drawing of $K_{2,n}$ in which both vertices on the 2-side have the same clockwise rotation. For two vertices $u,v$ on the…