Related papers: Matchings avoiding ordered patterns

We determine the maximum number of edges in a $K_4$-minor-free $n$-vertex graph of girth $g$, when $g = 5$ or $g$ is even. We argue that there are many different $n$-vertex extremal graphs, if $n$ is even and $g$ is odd.

We characterize the connected graphs of given order $n$ and given independence number $\alpha$ that maximize the number of maximum independent sets. For $3\leq \alpha\leq n/2$, there is a unique such graph that arises from the disjoint…

For a sequence $(H_i)_{i=1}^k$ of graphs, let $\textrm{nim}(n;H_1,\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$.…

Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem, maximizing the number of cliques of a fixed order in a graph with fixed number of vertices and…

This article provides sharp bounds for the maximum number of edges possible in a simple graph with restricted values of two of the three parameters, namely, maxi- mum matching size, independence number and maximum degree. We also construct…

A non-aligned drawing of a graph is a drawing where no two vertices are in the same row or column. Auber et al. showed that not all planar graphs have non-aligned drawings that are straight-line, planar, and in the minimal-possible $n\times…

A star-simple drawing of a graph is a drawing in which adjacent edges do not cross. In contrast, there is no restriction on the number of crossings between two independent edges. When allowing empty lenses (a face in the arrangement induced…

A drawing of a graph in the plane is a thrackle if every pair of edges intersects exactly once, either at a common vertex or at a proper crossing. Conway's conjecture states that a thrackle has at most as many edges as vertices. In this…

Let $G=(V,E)$ be an undirected graph without loops and multiple edges. A subset $C\subseteq V$ is called \emph{identifying} if for every vertex $x\in V$ the intersection of $C$ and the closed neighbourhood of $x$ is nonempty, and these…

An ordered graph $G$ is a graph whose vertex set is a subset of integers. The edges are interpreted as tuples $(u,v)$ with $u < v$. For a positive integer $s$, a matrix $M \in \mathbb{Z}^{s \times 4}$, and a vector $\mathbf{p} =…

A $k$-ranking is a vertex $k$-coloring such that if two vertices have the same color any path connecting them contains a vertex of larger color. The rank number of a graph is smallest $k$ such that $G$ has a $k$-ranking. For certain graphs…

A topological graph is a graph drawn in the plane. A topological graph is $k$-plane, $k>0$, if each edge is crossed at most $k$ times. We study the problem of partitioning the edges of a $k$-plane graph such that each partite set forms a…

An isolating set in a graph is a set $X$ of vertices such that every edge of the graph is incident with a vertex of $X$ or its neighborhood. The isolation number of a graph, or equivalently the vertex-edge domination number, is the minimum…

The Tur\'{a}n number of a graph $H$, $ex(n,H)$, is the maximum number of edges in a simple graph of order $n$ which does not contain $H$ as a subgraph. Let $k\cdot P_3$ denote $k$ disjoint copies of a path on $3$ vertices. In this paper, we…

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $\delta(H)>0$ and $\Delta(H)$ denote the minimum degree and maximum degree of $H$,…

For a set of graphs $\mathcal{F}$, the extremal number $ex(n;\mathcal{F})$ is the maximum number of edges in a graph of order $n$ not containing any subgraph isomorphic to some graph in $\mathcal{F}$. If $\mathcal{F}$ contains a graph on…

Burr and Erd\H{o}s conjectured in 1976 that for every two integers $k>\ell\geqslant 0$ satisfying that $k\mathbb{Z}+\ell$ contains an even integer, an $n$-vertex graph containing no cycles of length $\ell$ modulo $k$ can contain at most a…

How many matchings on the vertex set V={1,2,...,2n} avoid a given configuration of three edges? Chen, Deng and Du have shown that the number of matchings that avoid three nesting edges is equal to the number of matchings avoiding three…

Extremal problems related to the enumeration of graph substructures, such as independent sets, matchings, and induced matchings, have become a prominent area of research with the advancement of graph theory. A subset of vertices is called a…

Let $k\ge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $\binom{n-1}{k-1} + \lfloor\frac{n-1}{k}…