Related papers: Sidon Sets and graphs without 4-cycles

There are two particular $\Theta_6$-graphs - the 6-cycle graphs with a diagonal. We find the planar Tur\'an number of each of them, i.e. the maximum number of edges in a planar graph $G$ of $n$ vertices not containing the given $\Theta_6$…

In this paper we estimate the planar Tur\'an number $\mathrm{ex}_\mathcal{P}(n,H)$ of some graphs $H$, i.e., the maximum number of edges in a planar graph $G$ of $n$ vertices not containing $H$ as a subgraph. We give a new, short proof when…

The generalized Tur\'{a}n number ${\rm ex}(G,H)$ is the maximum number of edges in an $H$-free subgraph of a graph $G.$ It is an important extension of the classical Tur\'{a}n number ${\rm ex}(n,H)$, which is the maximum number of edges in…

The Tur\'{a}n number of a graph $H$, denoted $\mbox{ex}(n,H)$, is the maximum number of edges in an $n$-vertex graph with no subgraph isomorphic to $H$. Solymosi conjectured that if $H$ is any graph and $\mbox{ex}(n,H) = O(n^{\alpha})$…

We show that, for each fixed $k$, an $n$-vertex graph not containing a cycle of length $2k$ has at most $80\sqrt{k}\log k\cdot n^{1+1/k}+O(n)$ edges.

We show that every $r$-uniform hypergraph on $n$ vertices which does not contain a tight cycle has at most $O(n^{r-1} (\log n)^5)$ edges. This is an improvement on the previously best-known bound, of $n^{r-1} e^{O(\sqrt{\log n})}$, due to…

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…

Let $G$ be an $n$-vertex graph obtained by adding chords to a cycle of length $n$. Markstr\"{o}m asked for the maximum number of edges in $G$ if there are no two cycles in $G$ with the same length. A simple counting argument shows that such…

For a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n,\mathcal{F})$ is the maximum number of edges in an $n$-vertex graph containing no member of $\mathcal{F}$ as a subgraph. The maximum number of edges in an $n$-vertex connected…

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…

We give a simple geometric interpretation of an algebraic construction of Wenger that yields $n$-vertex graphs with no cycle of length $4$, $6$ or $10$ and close to the maximum number of edges.

We study the Tur\'{a}n problem for highly symmetric bipartite graphs arising from geometric shapes and periodic tilings commonly found in nature. 1. The prism $C_{2\ell}^{\square}:=C_{2\ell}\square K_{2}$ is the graph consisting of two…

In the 1960s, Erd\H{o}s and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by…

Given a fixed graph H, we say that a graph G is H-free if G does not contain H as a subgraph. The Tur\'an number ex(n, H) of H is the maximum number of edges in an n-vertex H-free graph. The study of Tur\'an number of graphs is a central…

We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\geq 4$. For $k\geq 6$ our constructions have $> (\frac{1}{4} -\varepsilon)n^2$ edges, which is asymptotically best possible by Tur\'an's…

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.

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,…

For a graph $H$, the Tur\'{a}n number of $H$, denoted by ex$(n,H)$, is the maximum number of edges of an $n$-vertex $H$-free graph. Let $g(n,H)$ denote the maximum number of edges not contained in any monochromatic copy of $H$ in a…

In 2016, Dowden initiated the study of planar Tur\'an-type problems, which has since attracted considerable attention. Recently, Bekos et al. proved that every $K_3$-free $1$-planar graph on $n\ge 4$ vertices has at most $3n-6$ edges. In…

In 1975, P. Erd\"{o}s proposed the problem of determining the maximum number $f(n)$ of edges in a graph of $n$ vertices in which any two cycles are of different lengths. In this paper, it is proved that $$f(n)\geq n+32t-1$$ for…