Turán's Theorem says that an extremal Kr+1-free graph is r-partite. The Stability Theorem of Erdős and Simonovits shows that if a Kr+1-free graph with n vertices has close to the maximal tr(n) edges, then it is close to being r-partite. In this paper we determine exactly the Kr+1-free graphs with at least m edges that are farthest from being r-partite, for any m≥tr(n)−δrn2. This extends work by Erdős, Győri and Simonovits, and proves a conjecture of Balogh, Clemen, Lavrov, Lidický and Pfender.
The Turán 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 Turán's theorem. However, when $r\ge 3$, the problem remains open. We model the problem as an integer program and call the underlying polytope the Turán polytope. We draw parallels between the latter and the stable set polytope: we show that generalized and transformed versions of the web and wheel inequalities are also facet-defining for the Turán polytope. We also show that clique inequalities and what we call doubling inequalities are facet-defining when $r=2$. These facets lead to a simple new polyhedral proof of Turán's theorem.
Let $G = (V, E)$ be a graph and $k \geq 0$ an integer. A $k$-independent set $S \subseteq G$ is a set of vertices such that the maximum degree in the graph induced by $S$ is at most $k$. Denote by $\alpha_{k}(G)$ the maximum cardinality of a $k$-independent set of $G$. For a graph $G$ on $n$ vertices and average degree $d$, Turán's theorem asserts that $\alpha_{0}(G) \geq \frac{n}{d+1}$, where the equality holds if and only if $G$ is a union of cliques of equal size. For general $k$ we prove that $\alpha_{k}(G) \geq \dfrac{(k+1)n}{d+k+1}$, improving on the previous best bound $\alpha_{k}(G) \geq \dfrac{(k+1)n}{ \lceil d \rceil+k+1}$ of Caro and Hansberg [E-JC, 2013]. For $1$-independence we prove that equality holds if and only if $G$ is either an independent set or a union of almost-cliques of equal size (an almost-clique is a clique on an even number of vertices minus a $1$-factor). For $2$-independence, we prove that equality holds if and only if $G$ is an independent set. Furthermore when $d>0$ is an integer divisible by 3 we prove that $\alpha_2(G) \geq \dfrac{3n}{d+3} \left( 1 + \dfrac{12}{5d^2 + 25d + 18} \right)$.
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 Turán graph turns out to be the unique extremal graph. For $(r-1)|M|<n$, there is a whole family of extremal graphs, which we describe explicitly. In addition we provide corresponding stability results.
Exact stability for Turán’s Theorem
