Related papers: Extremal problems on planar graphs without k edge-…
The problem of determining the maximum number of edges in an $n$-vertex graph that does not contain a 4-cycle has a rich history in extremal graph theory. Using Sidon sets constructed by Bose and Chowla, for each odd prime power $q$ we…
The spread of a graph $G$ is the difference between the largest and smallest eigenvalue of the adjacency matrix of $G$. Gotshall, O'Brien and Tait conjectured that for sufficiently large $n$, the $n$-vertex outerplanar graph with maximum…
A 1-planar graph refers to a graph that can be drawn on the plane such that each edge has at most one crossing. In this paper, focusing on the spectral Tur\'{a}n-type problems of $1$-planar graphs, we determine completely the unique…
While investigating odd-cycle free hypergraphs, Gy\H{o}ri and Lemons introduced a colored version of the classical theorem of Erd\H{o}s and Gallai on $P_k$-free graphs. They proved that any graph $G$ with a proper vertex coloring and no…
P. Erd\H{o}s, J. Pach, R. Pollack, and Z. Tuza [J. Combin. Theory, B 47 (1989), 279--285] made conjectures for the maximum diameter of connected graphs without a complete subgraph $K_{k+1}$, which have order $n$ and minimum degree $\delta$.…
Given a family $\mathcal{F}$, a graph is $\mathcal{F}$-free if it does not contain any graph in $\mathcal{F}$ as a subgraph. We study the topic of "extremal" planar graphs initiated by Dowden [J. Graph Theory 83 (2016) 213--230], that is,…
In 1963, Corr\'adi and Hajnal proved that for all $k\geq1$ and $n\geq3k$, every graph $G$ on $n$ vertices with minimum degree $\delta(G)\geq2k$ contains $k$ disjoint cycles. The bound $\delta(G) \geq 2k$ is sharp. Here we characterize those…
In this paper, we make progress on a question related to one of Galvin that has attracted substantial attention recently. The question is that of determining among all graphs $G$ with $n$ vertices and $\Delta(G)\leq r$, which has the most…
In 1975, Erd\H{o}s and Sauer asked to estimate, for any constant $r$, the maximum number of edges an $n$-vertex graph can have without containing an $r$-regular subgraph. In a recent breakthrough, Janzer and Sudakov proved that any…
The spectral radius and rank of a graph are defined to be the spectral radius and rank of its adjacency matrix, respectively. It is an important problem in spectral extremal graph theory to determine the extremal graph that has the maximum…
More than forty years ago, Erd\H{o}s conjectured that for any T <= N/K, every K-uniform hypergraph on N vertices without T disjoint edges has at most max{\binom{KT-1}{K}, \binom{N}{K} - \binom{N-T+1}{K}} edges. Although this appears to be a…
A graph whose vertices are points in the plane and whose edges are noncrossing straight-line segments of unit length is called a \emph{matchstick graph}. We prove two somewhat counterintuitive results concerning the maximum number of edges…
Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well studied function, the planar…
A drawing of a graph is $k$-plane if every edge contains at most $k$ crossings. A $k$-plane drawing is saturated if we cannot add any edge so that the drawing remains $k$-plane. It is well-known that saturated $0$-plane drawings, that is,…
Let $\mc{F}$ be a family of graphs. A graph is {\em $\mc{F}$-free} if it contains no copy of a graph in $\mc{F}$ as a subgraph. A cornerstone of extremal graph theory is the study of the {\em Tur\'an number} $ex(n,\mc{F})$, the maximum…
It is shown that for a constant $t\in \mathbb{N}$, every simple topological graph on $n$ vertices has $O(n)$ edges if it has no two sets of $t$ edges such that every edge in one set is disjoint from all edges of the other set (i.e., the…
A well-known theorem of Erd\H{o}s and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Gy\H{o}ri, Katona and Lemons gave an extension of this result to hypergraphs by determining…
We show the quarter of a century old conjecture that every $K_4$-free graph with $n$ vertices and $\lfloor n^2/4 \rfloor +k$ edges contains $k$ pairwise edge disjoint triangles.
We prove that, for the edge ideal of a graph whose cycles are pairwise vertex-disjoint, the arithmetical rank is bounded above by the sum of the number of cycles and the maximum height of its associated primes.
Paul Erd\H{o}s suggested the following problem: Determine or estimate the number of maximal triangle-free graphs on $n$ vertices. Here we show that the number of maximal triangle-free graphs is at most $2^{n^2/8+o(n^2)}$, which matches the…