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Register a new userSpectral saturation: inverting the spectral Turan theorem
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Vladimir Nikiforov
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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 a complete r-partite graph of size log n with one edge added to the first part. These results complete a project of Erdos from 1963. We also give corresponding stability results.
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Classical questions in extremal graph theory concern the asymptotics of $operatorname{ex}(G, mathcal{H})$ where $mathcal{H}$ is a fixed family of graphs and $G=G_n$ is taken from a `standard increasing sequence of host graphs $(G_1, G_2, dots)$, most often $K_n$ or $K_{n,n}$. Inverting the question, we can instead ask how large $e(G)$ can be with respect to $operatorname{ex}(G,mathcal{H})$. We show that the standard sequences indeed maximize $e(G)$ for some choices of $mathcal{H}$, but not for others. Many interesting questions and previous results arise very naturally in this context, which also, unusually, gives rise to sensible extremal questions concerning multigraphs and non-uniform hypergraphs.
We give a bound on the spectral radius of a graph implying a quantitative version of the Erdos-Stone theorem.
Given $r$-uniform hypergraphs $G$ and $H$ the Turan number $rm ex(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $rm ex(G, H)$ when $G=G_{n,p}^{(r)}$, the ErdH{o}s-Renyi random $r$-uniform hypergraph, and $H=C_{2ell}^{(r)}$, the $r$-uniform linear cycle of length $2ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine $rm ex(G_{n,p}^{(r)}, C_{2ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $ell$ grows. Our main technical contribution is a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the supersaturation result depends on the codegree of some pairs of vertices in the underlying hypergraph. This approach could be used to prove similar results for other hypergraphs $H$.
We extend the classical stability theorem of Erdos and Simonovits in two directions: first, we allow the order of the forbidden graph to grow as log of order of the host graph, and second, our extremal condition is on the spectral radius of the host graph.
The Turan number of a graph $H$, denoted by $ex(n,H)$, is the maximum number of edges in any graph on $n$ vertices which does not contain $H$ as a subgraph. Let $P_{k}$ denote the path on $k$ vertices and let $mP_{k}$ denote $m$ disjoint copies of $P_{k}$. Bushaw and Kettle [Tur{a}n numbers of multiple paths and equibipartite forests, Combin. Probab. Comput. 20(2011) 837--853] determined the exact value of $ex(n,kP_ell)$ for large values of $n$. Yuan and Zhang [The Tur{a}n number of disjoint copies of paths, Discrete Math. 340(2)(2017) 132--139] completely determined the value of $ex(n,kP_3)$ for all $n$, and also determined $ex(n,F_m)$, where $F_m$ is the disjoint union of $m$ paths containing at most one odd path. They also determined the exact value of $ex(n,P_3cup P_{2ell+1})$ for $ngeq 2ell+4$. Recently, Bielak and Kieliszek [The Tur{a}n number of the graph $2P_5$, Discuss. Math. Graph Theory 36(2016) 683--694], Yuan and Zhang [Tur{a}n numbers for disjoint paths, arXiv: 1611.00981v1] independently determined the exact value of $ex(n,2P_5)$. In this paper, we show that $ex(n,2P_{7})=max{[n,14,7],5n-14}$ for all $n ge 14$, where $[n,14,7]=(5n+91+r(r-6))/2$, $n-13equiv r,(text{mod }6)$ and $0leq r< 6$.
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