Jian Wang, Shipeng Wang, Weihua Yang, Xiaoli Yuan
For any positive integer $k$, we show that every maximal $C_{2k+1}$-free graph with at least $n^2/4-o(n^{3/2})$ edges contains an induced complete bipartite subgraph on $(1-o(1))n$ vertices. We also show that this is best possible.
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Answering questions of Y. Rabinovich, we prove "stability" versions of upper bounds on maximal independent set counts in graphs under various restrictions. Roughly these say that being close to the maximum implies existence of a large induced matching or triangle matching (depending on assumptions).
Popielarz, Sahasrabudhe and Snyder in 2018 proved that maximal $K_{r+1}$-free graphs with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contain a complete $r$-partite subgraph on $n-o(n)$ vertices. This was very recently extended to odd cycles in place of $K_3$ by Wang, Wang, Yang and Yuan. We further extend it to some other 3-chromatic graphs, and obtain some other stability results along the way.
For $r \geq 2$, we show that every maximal $K_{r+1}$-free graph $G$ on $n$ vertices with $(1-\frac{1}{r})\frac{n^2}{2}-o(n^{\frac{r+1}{r}})$ edges contains a complete $r$-partite subgraph on $(1 - o(1))n$ vertices. We also show that this is best possible. This result answers a question of Tyomkyn and Uzzell.
Let $F$ be an $(r+1)$-color critical graph with $r\geq 2$, that is, $χ(F)=r+1$ and there is an edge $e$ in $F$ such that $χ(F-e)=r$. Gerbner recently conjectured that every $n$-vertex maximal $F$-free graph with at least $(1-\frac{1}{r})\frac{n^2}{2}- o(n^{\frac{r+1}{r}})$ edges contains an induced complete $r$-partite graph on $n-o(n)$ vertices. Let $F_{s,k}$ be a graph obtained from $s$ copies of $C_{2k+1}$ by sharing a common edge. In this paper, we show that for all $k\geq 2$ if $G$ is an $n$-vertex maximal $F_{s,k}$-free graph with at least $n^{2}/4 - o(n^{\frac{s+2}{s+1}})$ edges, then $…
We prove that if $M$ is a maximal $k$-edge-colorable subgraph of a multigraph $G$ and if $F = \{v \in V(G) : d_M(v) \leq k-μ(v)\}$, then $d_F(v) \leq d_M(v)$ for all $v \in F$. (When $G$ is a simple graph, the set $F$ is just the set of vertices having degree less than $k$ in $M$.) This implies Vizing's Theorem as well as a special case of Tuza's Conjecture on packing and covering of triangles. A more detailed version of our result also implies Vizing's Adjacency Lemma for simple graphs.
If K is an odd-dimensional flag closed manifold, flag generalized homology sphere or a more general flag weak pseudomanifold with sufficiently many vertices, then the maximal number of edges in K is achieved by the balanced join of cycles. The proof relies on stability results from extremal graph theory. In the case of manifolds we also offer an alternative (very) short proof utilizing the non-embeddability theorem of Flores. The main theorem can also be interpreted without the topological contents as a graph-theoretic extremal result about a class of graphs such that 1) every maximal clique i…
Combining Pósa's rotation lemma with a technique of Kopylov in a novel approach, we prove a generalization of the Erdős-Gallai theorems on cycles and paths. This implies a clique version of the Erdős-Gallai stability theorems and also provides alternative proofs for some recent results.
We prove a common generalization of the maximal independent arborescence packing theorem of Cs. Király and two of our earlier works about packing branchings in infinite digraphs.
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.
A stability theorem for multi-partite graphs
本文提出多部图的稳定性定理，解决了关于不含顶点不相交克lique副本的多部图最大边数的猜想。
Both the Simonovits stability theorem and the Nikiforov spectral stability theorem are powerful tools for solving exact values of Turán numbers in extremal graph theory. Recently, Füredi [J. Combin. Theory Ser. B 115 (2015)] provided a concise and contemporary proof of the Simonovits stability theorem. In this note, we present a unified treatment for some extremal graph problems, including short proofs of Nikiforov's spectral stability theorem and the clique stability theorem proved recently by Ma and Qiu [European J. Combin. 84 (2020)]. Moreover, some spectral extremal problems related to the…
A recent paper of Church, Ellenberg, and Farb uses topology and representation theory of the symmetric group to prove enumerative results about square-free polynomials and F-stable maximal tori of the general linear group over the algebraic closure of F_q. In this note, we use generating functions to give elementary proofs of some of their results, and some extensions.
Using the polynomial method in additive number theory, this article establishes a new addition theorem for the set of subsums of a set satisfying $A\cap(-A)=\emptyset$ in $\mathbb{Z}/p\mathbb{Z}$: \[|Σ(A)|\geqslant\min{p,1+\frac{|A|(|A|+1)}{2}}.\] The proof is similar in nature to Alon, Nathanson and Ruzsa's proof of the Erdös-Heilbronn conjecture (proved initially by Dias da Silva and Hamidoune \cite{DH}). A key point in the proof of this theorem is the evaluation of some binomial determinants that have been studied in the work of Gessel and Viennot. A generalization to the set of subsums of …
Let $K$ be a maximal lattice-free set in $\mathbb{R}^d$, that is, $K$ is convex and closed subset of $\mathbb{R}^d$, the interior of $K$ does not cointain points of $\mathbb{Z}^d$ and $K$ is inclusion-maximal with respect to the above properties. A result of Lovász assert that if $K$ is $d$-dimensional, then $K$ is a polyhedron with at most $2^d$ facets, and the recession cone of $K$ is spanned by vectors from $\mathbb{Z}^d$. A first complete proof of mentioned Lovász's result has been published in a paper of Basu, Conforti, Cornuéjols and Zambelli (where the authors use Dirichlet's approximat…
Stability of maximal relative projection constants
该论文针对$\ell_\infty^n$子空间的极大相对投影常数稳定性问题，证明当$n\ge 2^r\binom{r+1}{2}$时$\lambda(r,n)$稳定于$\lambda(r)$，解答了Basso的问题。
A family of sets is called $r$-\emph{cover free} if no set in the family is contained in the union of $r$ (or less) other sets in the family. A $1$-cover free family is simply an antichain with respect to set inclusion. Thus, Sperner's classical result determines the maximal cardinality of a $1$-cover free family of subsets of an $n$-element set. Estimating the maximal cardinality of an $r$-cover free family of subsets of an $n$-element set for $r>1$ was also studied. In this note we are interested in the following probabilistic variant of this problem. Let $S_0,S_1,\ldots, S_r$ be independent…
A result of Lehrer describes a beautiful relationship between topological and combinatorial data on certain families of varieties with actions of finite reflection groups. His formula relates the cohomology of complex varieties to point counts on associated varieties over finite fields. Church, Ellenberg, and Farb use their representation stability results on the cohomology of flag manifolds, together with classical results on the cohomology rings, to prove asymptotic stability for "polynomial" statistics on associated varieties over finite fields. In this paper we investigate the underlying a…
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