Related papers: Turan's theorem inverted

We prove that if the spectral radius of a graph G of order n is larger than the spectral radius of the r-partite Turan graph of the same order, then G contains various supergraphs of the complete graph of order r+1. In particular G contains…

For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Tur\'an's Theorem says that equality holds for complete graphs.…

Tur\'an's Theorem says that an extremal $K_{r+1}$-free graph is $r$-partite. The Stability Theorem of Erd\H{o}s and Simonovits shows that if a $K_{r+1}$-free graph with $n$ vertices has close to the maximal $t_r(n)$ edges, then it is close…

We determine the maximum number of edges of an $n$-vertex graph $G$ with the property that none of its $r$-cliques intersects a fixed set $M\subset V(G)$. For $(r-1)|M|\ge n$, the $(r-1)$-partite Turan graph turns out to be the unique…

We propose a strengthening of the conclusion in Tur\'an's (3,4)-conjecture in terms of algebraic shifting, and show that its analogue for graphs does hold. In another direction, we generalize the Mantel-Tur\'an theorem by weakening its…

The Tur\'an hypergraph problem asks to find the maximum number of $r$-edges in a $r$-uniform hypergraph on $n$ vertices that does not contain a clique of size $a$. When $r=2$, i.e., for graphs, the answer is well-known and can be found in…

Confirming a conjecture of Vera T. S\'os in a very strong sense, we give a complete solution to Tur\'an's hypergraph problem for the Fano plane. That is we prove for $n\ge 8$ that among all $3$-uniform hypergraphs on $n$ vertices not…

We show that if G is a graph of sufficiently large order n containing as many r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the…

A conjecture of Erd\H{o}s from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…

Hajos' conjecture that every simple even graph on $n$ vertices can be decomposed into at most $(n-1)/2$ cycles (see L. Lovasz, On covering of graphs, in: P. Erdos, G.O.H. Katona (Eds.), Theory of Graphs, Academic Press, New York, 1968, pp.…

Let $H$ be a graph. We show that if $r$ is large enough as a function of $H$, then the $r$-partite Tur\'an graph maximizes the number of copies of $H$ among all $K_{r+1}$-free graphs on a given number of vertices. This confirms a conjecture…

The extension of an $r$-uniform hypergraph $G$ is obtained from it by adding for every pair of vertices of $G$, which is not covered by an edge in $G$, an extra edge containing this pair and $(r-2)$ new vertices. In this paper we determine…

Let $\mathcal{F}$ be a family of $r$-graphs. The Tur\'an number $ex_r(n;\mathcal{F})$ is defined to be the maximum number of edges in an $r$-graph of order $n$ that is $\mathcal{F}$-free. The famous Erd\H{o}s Matching Conjecture shows that…

In 1969 Erdoes found a lower bound on the number of (r+1)-cliques sharing an edge in graphs with n vertices and t(r,n)+1 edges, where t(r,n) is the size of the Turan graph of order n and r color classes. We improve Erdoes's bound and prove…

In 1952, Dirac proved that every $2$-connected $n$-vertex graph with the minimum degree $k+1$ contains a cycle of length at least $\min\{n, 2(k+1)\}$. Here we obtain a stability version of this result by characterizing those graphs with…

The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines the exact bound on the number of edges $t_r(n)$ in an $n$-vertex graph that does not contain a clique of size $r+1$. We establish an interesting link between…

Turan's Theorem states that every graph of a certain edge density contains a complete graph $K^k$ and describes the unique extremal graphs. We give a similar Theorem for l-partite graphs. For large l, we find the minimal edge density…

An ordered graph $H$ is a simple graph with a linear order on its vertex set. The corresponding Tur\'an problem, first studied by Pach and Tardos, asks for the maximum number $\text{ex}_<(n,H)$ of edges in an ordered graph on $n$ vertices…

As a variant of the famous Tur\'an problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of…