Combinatorics (math.CO)

Abstract: First, we consider order-$n$ ribbon tilings of an $M$-by-$N$ rectangle $R_{M,N}$ where $M$ and $N$ are much larger than $n$. We prove the existence of the growth rate $\gamma_n$ of the number of tilings and show that $\gamma_n \leq (n-1) \ln 2$. Then, we study a rectangle $R_{M,N}$ with fixed width $M=n$, called a strip. We derive lower and upper bounds on the growth rate $\mu_n$ for strips as $ \ln n - 1 + o(1) \leq \mu_n \leq \ln n $. Besides, we construct a recursive system which enables us to enumerate the order-$n$ ribbon tilings of a strip for all $n \leq 8$ and calculate the corresponding generating functions.

Abstract: A well-known theorem of R\"odl says that for every graph $H$, and every $\epsilon>0$, there exists $\delta>0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge\delta|G|$ such that one of $G[X],\overline{G}[X]$ has edge-density at most $\epsilon$. But how does $\delta$ depend on $\epsilon$? Fox and Sudakov conjectured that the dependence can be taken to be polynomial: for all $H$ there exists $c>0$ such that for all $\epsilon\in(0,1/2)$, R\"odl's theorem holds with $\delta=\epsilon^c$. This conjecture implies the Erd\H{o}s-Hajnal conjecture, and until now it had not been verified for any non-trivial graphs $H$. Our first result shows that it is true when $H=P_4$ (indeed, we can take $\delta=\epsilon$, and insist that one of $G[X],\overline{G}[X]$ has maximum degree at most $\epsilon^2|G|$).) We will generalize this, and to do so, we need to work with an even stronger property. Let us say $H$ is {\em viral} if there exists $c>0$ such that for all $\epsilon\in(0,1/2)$, if $G$ contains at most $\epsilon^c|G|^{|H|}$ induced copies of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge \epsilon^c|G|$ such that one of $G[X],\overline{G}[X]$ has density at most $\epsilon$. We will show that $P_4$ is viral, using a ``polynomial $P_4$-removal lemma'' of Alon and Fox. We will also show that viral graphs are closed under vertex-substitution, and so all graphs that can be obtained by substitution from copies of $P_4$ are viral. (In a subsequent paper, it will be shown that all graphs currently known to satisfy the Erd\H{o}s-Hajnal conjecture are in fact viral.) Finally, we give a different strengthening of R\"odl's theorem: we show that if $G$ does not contain an induced copy of $P_4$, then its vertices can be partitioned into at most $480\epsilon^{-4}$ subsets $X$ such that one of $G[X],\overline{G}[X]$ has density at most $\epsilon$.

Abstract: It is well-known that every tournament contains a Hamilton path, and every strongly connected tournament contains a Hamilton cycle. This paper establishes transversal generalizations of these classical results. For a collection $\mathbf{T}=\{T_1,\dots,T_m\}$ of not-necessarily distinct tournaments on a common vertex set $V$, an $m$-edge directed graph $\mathcal{D}$ with vertices in $V$ is called a $\mathbf{T}$-transversal if there exists a bijection $\phi\colon E(\mathcal{D})\to [m]$ such that $e\in E(T_{\phi(e)})$ for all $e\in E(\mathcal{D})$. We prove that for sufficiently large $m$ with $m=|V|-1$, there exists a $\mathbf{T}$-transversal Hamilton path. Moreover, if $m=|V|$ and at least $m-1$ of the tournaments $T_1,\ldots,T_m$ are assumed to be strongly connected, then there is a $\mathbf{T}$-transversal Hamilton cycle. In our proof, we utilize a novel way of partitioning tournaments which we dub $\mathbf{H}$-partition.

Abstract: The distance spectrum of a connected graph is the spectrum of its distance matrix. We characterize those bicyclic graphs and split graphs whose second largest distance eigenvalues belong to the interval $(-\infty,-\frac{1}{2})$. Any such graph is chordal.

Abstract: The Sombor index $SO(G)$ of a graph $G$ is the sum of the edge weights $\sqrt{d^2_G(u)+d^2_G(v)}$ of all edges $uv$ of $G$, where $d_G(u)$ denotes the degree of the vertex $u$ in $G$. A connected graph $G = (V ,E)$ is called a quasi-tree, if there exists $u\in V (G)$ such that $G-u$ is a tree. Denote $\mathscr{Q}(n,k)$=\{$G$: $G$ is a quasi-tree graph of order $n$ with $G-u$ being a tree and $d_G(u)=k$\}. In this paper, we determined the maximum, the second maximum and the third maximum Sombor indices of all quasi-tree graphs in $\mathscr{Q}(n,k)$, respectively. Moreover, we characterized their corresponding extremal graphs, respectively.

Abstract: We present a new combinatorial and conjectural algorithm for computing the Mullineux involution for the symmetric group and its Hecke algebra. This algorithm is built on a conjectural property of crystal isomorphisms which can be rephrased in a purely combinatorial way.

Abstract: The notion of combinatorially rich set was first introduced by V. Bergelson and D. Glasscock. The goal of this paper is to study these subsets in the setting of adequate partial semigroups and describes its connection with the other concepts of large sets.

Abstract: Motivated by Hadwiger's conjecture, we study the problem of finding the densest possible $t$-vertex minor in graphs of average degree at least $t-1$. We show that if $G$ has average degree at least $t-1$, it contains a minor on $t$ vertices with at least $(\sqrt{2}-1-o(1))\binom{t}{2}$ edges. We show that this cannot be improved beyond $\left(\frac{3}{4}+o(1)\right)\binom{t}{2}$. Finally, for $t\leq 6$ we exactly determine the number of edges we are guaranteed to find in the densest $t$-vertex minor in graphs of average degree at least $t-1$.

Abstract: Recently, Alon introduced the notion of an $H$-code for a graph $H$: a collection of graphs on vertex set $[n]$ is an $H$-code if it contains no two members whose symmetric difference is isomorphic to $H$. Let $D_{H}(n)$ denote the maximum possible cardinality of an $H$-code, and let $d_{H}(n)=D_{H}(n)/2^{n \choose 2}$. Alon observed that a lower bound on $d_{H}(n)$ can be obtained by attaining an upper bound on the number of colors needed to edge-color $K_n$ so that every copy of $H$ has an odd color class. Motivated by this observation, we define $g(G,H)$ to be the minimum number of colors needed to edge-color a graph $G$ so that every copy of $H$ has an odd color class. We prove $g(K_n,K_5) \le n^{o(1)}$ and $g(K_{n,n}, C_4) \le \frac{t}{2(t-1)}n+o(n)$ for all odd integers $t\geq 5$. The first result shows $d_{K_5}(n) \ge \frac{1}{n^{o(1)}}$ and was obtained independently in arXiv:2306.14682.

Abstract: Given a nonnegative integer weight $f(v)$ for each vertex $v$ in a multigraph $G$, an {\it $f$-bounded subgraph} of $G$ is a multigraph $H$ contained in $G$ such that $d_H(v)\le f(v)$ for all $v\in V(G)$. Using Tutte's $f$-Factor Theorem, we give a new proof of the min-max relation for the maximum size of an $f$-bounded subgraph of $G$. When $f(v)=1$ for all $v$, the formula reduces to the classical Tutte--Berge Formula for the maximum size of a matching.

Abstract: For a function $f$ on the hypercube $\{0,1\}^n$ with Fourier expansion $f=\sum_{S\subseteq[n]}\hat f(S)\chi_S$, the hypercontractive inequality for the noise operator allows bounding norms of $T_\rho f=\sum_S\rho^{|S|}\hat f(S)\chi_S$ in terms of norms of $f$. If $f$ is Boolean-valued, the level-$d$ inequality allows bounding the norm of $f^{=d}=\sum_{|S|=d}\hat f(S)\chi_S$ in terms of $E[f]$. These two inequalities play a central role in analysis of Boolean functions and its applications. While both inequalities hold in a sharp form when the hypercube is endowed with the uniform measure, they do not hold for more general discrete product spaces. Finding a `natural' generalization was a long-standing open problem. In [P. Keevash et al., Global hypercontractivity and its applications, J. Amer. Math. Soc., to appear], a hypercontractive inequality for this setting was presented, that holds for functions which are `global' -- namely, are not significantly affected by a restriction of a small set of coordinates. This hypercontractive inequality is not sharp, which precludes applications to the symmetric group $S_n$ and to other settings where sharpness of the bound is crucial. Also, no level-$d$ inequality for global functions over general discrete product spaces is known. We obtain sharp versions of the hypercontractive inequality and of the level-$d$ inequality for this setting. Our inequalities open the way for diverse applications to extremal set theory and to group theory. We demonstrate this by proving quantitative bounds on the size of intersecting families of sets and vectors under weak symmetry conditions and by describing numerous applications to the study of functions on $S_n$ -- including hypercontractivity and level-$d$ inequalities, character bounds, variants of Roth's theorem and of Bogolyubov's lemma, and diameter bounds, that were obtained using our techniques.

Abstract: A sunflower with $r$ petals is a collection of $r$ sets over a ground set $X$ such that every element in $X$ is in no set, every set, or exactly one set. Erd\H{o}s and Rado \cite{er} showed that a family of sets of size $n$ contains a sunflower if there are more than $n!(r-1)^n$ sets in the family. Alweiss et al. \cite{alwz} and subsequently, Rao \cite{rao} improved this bound to $(O(r \log(rn))^n$. We study the case where the pairwise intersections of the set family are restricted. In particular, we improve the best-known bound for set families when the size of the pairwise intersections of any two sets is in a set $L$. We also present a new bound for the special case when the set $L$ is the nonnegative integers less than or equal to $d$ using the techniques of Alweiss et al. \cite{alwz}.