arXiv subjects

Sam Spiro

Publications and source records attributed to Sam Spiro.

53 records · Page 3Linked to original sources

arXiv2021

Relative Turán Numbers for Hypergraph Cycles

For an $r$-uniform hypergraph $H$ and a family of $r$-uniform hypergraphs $\mathcal{F}$, the relative Turán number $\mathrm{ex}(H,\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $H$. In this paper we give lower bounds on $\mathrm{ex}(H,\mathcal{F})$ for certain families of hypergraph cycles $\mathcal{F}$ such as Berge cycles and loose cycles. In particular, if $\mathcal{C}_\ell^3$ denotes the set of all $3$-uniform Berge $\ell$-cycles and $H$ is a 3-uniform hypergraph with maximum degree $Δ$, we prove \[\mathrm{ex}(H,\mathcal{C}_4^{3})\ge Δ^{-3/4-o(1)}e(H),\] \[\mathrm{ex}(H,\mathcal{C}_5^{3})\ge Δ^{-3/4-o(1)}e(H),\] and these bounds are tight up to the $o(1)$ term.

math.CO↗

arXiv2021

Online Card Games

Consider the following one player game. A deck containing $m$ copies of $n$ different card types is shuffled uniformly at random. Each round the player tries to guess the next card in the deck, and then the card is revealed and discarded. It was shown by Diaconis, Graham, He, and Spiro that if $m$ is fixed, then the maximum expected number of correct guesses that the player can achieve is asymptotic to $H_m \log n$, where $H_m$ is the $m$th harmonic number. In this paper we consider an adversarial version of this game where a second player shuffles the deck according to some (possibly non-uniform) distribution. We prove that a certain greedy strategy for the shuffler is the unique optimal strategy in this game, and that the guesser can achieve at most $\log n$ expected correct guesses asymptotically for fixed $m$ against this greedy strategy.

math.PR↗

arXiv2021

The Wiener Index of Signed Graphs

The Wiener index of a graph $W(G)$ is a well studied topological index for graphs. An outstanding problem of Šolt{é}s is to find graphs $G$ such that $W(G)=W(G-v)$ for all vertices $v\in V(G)$, with the only known example being $G=C_{11}$. We relax this problem by defining a notion of Wiener indices for signed graphs, which we denote by $W_σ(G)$, and under this relaxation we construct many signed graphs such that $W_σ(G)=W_σ(G-v)$ for all $v\in V(G)$. This ends up being related to a problem of independent interest, which asks when it is possible to $2$-color the edges of a graph $G$ such that there is a path between any two vertices of $G$ which uses each color the same number of times.

math.CO↗

arXiv2021

Relative Turán Problems for Uniform Hypergraphs

For two graphs $F$ and $H$, the relative Turán number $\mathrm{ex}(H,F)$ is the maximum number of edges in an $F$-free subgraph of $H$. Foucaud, Krivelevich, and Perarnau \cite{FKP} and Perarnau and Reed \cite{PR} studied these quantities as a function of the maximum degree of $H$. In this paper, we study a generalization for uniform hypergraphs. If $F$ is a complete $r$-partite $r$-uniform hypergraph with parts of sizes $s_1,s_2,\dots,s_r$ with each $s_{i + 1}$ sufficiently large relative to $s_i$, then with $1/β= \sum_{i = 2}^r \prod_{j = 1}^{i - 1} s_j$ we prove that for any $r$-uniform hypergraph $H$ with maximum degree $Δ$, \[\mathrm{ex}(H,F)\ge Δ^{-β- o(1)} \cdot e(H).\] This is tight as $Δ\rightarrow \infty$ up to the $o(1)$ term in the exponent, since we show there exists a $Δ$-regular $r$-graph $H$ such that $\mathrm{ex}(H,F)=O(Δ^{-β}) \cdot e(H)$. Similar tight results are obtained when $H$ is the random $n$-vertex $r$-graph $H_{n,p}^r$ with edge-probability $p$, extending results of Balogh and Samotij \cite{BS} and Morris and Saxton \cite{MS}.

math.CO↗

arXiv2020

Counting Hypergraphs with Large Girth

Morris and Saxton used the method of containers to bound the number of $n$-vertex graphs with $m$ edges containing no $\ell$-cycles, and hence graphs of girth more than $\ell$. We consider a generalization to $r$-uniform hypergraphs. The {\em girth} of a hypergraph $H$ is the minimum $\ell$ such that for some $F \subseteq H$, there exists a bijection $\phi : E(C_\ell) \to E(F)$ with $e\subseteq \phi(e)$ for all $e\in E(C_\ell)$. Letting $N_m^r(n,\ell)$ denote the number of $n$-vertex $r$-uniform hypergraphs with $m$ edges and girth larger than $\ell$ and defining $\lambda = \lceil (r - 2)/(\ell - 2)\rfloor$, we show \[ N_m^r(n,\ell) \leq N_m^2(n,\ell)^{r - 1 + \lambda}\] which is tight when $\ell - 2 $ divides $r - 2$ up to a $1 + o(1)$ term in the exponent. This result is used to address the extremal problem for subgraphs of girth more than $\ell$ in random $r$-uniform hypergraphs.

math.CO↗

arXiv2020

An Averaging Processes on Hypergraphs

Consider the following iterated process on a hypergraph $H$. Each vertex $v$ has an initial vertex weight. At each step, we uniformly at random select an edge $F$ in $H$, and for each vertex $v$ in $F$ we replace the weight of $v$ by the average value of the vertex weights over all vertices in $F$. This is a generalization of an interactive process on graphs, first proposed by Aldous and Lanoue. In this paper, we use the eigenvalues of a Laplacian for hypergraphs to bound the rate of convergence for the iterated averaging process.

math.PR↗

arXiv2020

Forbidding $K_{2,t}$ traces in triple systems

Let $H$ and $F$ be hypergraphs. We say $H$ contains $F$ as a trace if there exists some set $S \subseteq V(H)$ such that $H|_S:=\{E\cap S: E \in E(H)\}$ contains a subhypergraph isomorphic to $F$. In this paper we give an upper bound on the number of edges in a $3$-uniform hypergraph that does not contain $K_{2,t}$ as a trace when $t$ is large. In particular, we show that $ \lim_{t\to \infty}\lim_{n\to \infty} \frac{\mathrm{ex}(n, \mathrm{Tr}_3(K_{2,t}))}{t^{3/2}n^{3/2}} = \frac{1}{6}.$ Moreover, we show $\frac{1}{2} n^{3/2} + o(n^{3/2}) \leq \mathrm{ex}(n, \mathrm{Tr}_3(C_4)) \leq \frac{5}{6} n^{3/2} + o(n^{3/2})$.

math.CO↗

arXiv2020

Counting Labeled Threshold Graphs with Eulerian Numbers

A threshold graph is any graph which can be constructed from the empty graph by repeatedly adding a new vertex that is either adjacent to every vertex or to no vertices. The Eulerian number $\genfrac{\langle}{\rangle}{0pt}{}{n}{k}$ counts the number of permutations of size $n$ with exactly $k$ ascents. Implicitly Beissinger and Peled proved that the number of labeled threshold graphs on $n\ge 2$ vertices is \[\sum_{k=1}^{n-1}(n-k)\genfrac{\langle}{\rangle}{0pt}{}{n-1}{k-1}2^k.\] Their proof used generating functions. We give a direct combinatorial proof of this result.

math.CO↗

arXiv2020

Triangle-free Subgraphs of Hypergraphs

In this paper, we consider an analog of the well-studied extremal problem for triangle-free subgraphs of graphs for uniform hypergraphs. A loose triangle is a hypergraph $T$ consisting of three edges $e,f$ and $g$ such that $|e \cap f| = |f \cap g| = |g \cap e| = 1$ and $e \cap f \cap g = \emptyset$. We prove that if $H$ is an $n$-vertex $r$-uniform hypergraph with maximum degree $\triangle$, then as $\triangle \rightarrow \infty$, the number of edges in a densest $T$-free subhypergraph of $H$ is at least \[ \frac{e(H)}{\triangle^{\frac{r-2}{r-1} + o(1)}}.\] For $r = 3$, this is tight up to the $o(1)$ term in the exponent. We also show that if $H$ is a random $n$-vertex triple system with edge-probability $p$ such that $pn^3\rightarrow\infty$ as $n\rightarrow\infty$, then with high probability as $n \rightarrow \infty$, the number of edges in a densest $T$-free subhypergraph is \[ \min\Bigl\{(1-o(1))p{n\choose3},p^{\frac{1}{3}}n^{2-o(1)}\Bigr\}.\] We use the method of containers together with probabilistic methods and a connection to the extremal problem for arithmetic progressions of length three due to Ruzsa and Szemerédi.

math.CO↗

arXiv2019

Subset Parking Functions

A parking function $(c_1,\ldots,c_n)$ can be viewed as having $n$ cars trying to park on a one-way street with $n$ parking spots, where car $i$ tries to park in spot $c_i$, and otherwise he parks in the leftmost available spot after $c_i$. Another way to view this is that each car has a set $C_i$ of "acceptable" parking spots, namely $C_i=[c_i,n]$, and that each car tries to park in the leftmost available spot that they find acceptable. Motivated by this, we define a subset parking function $(C_1,\ldots,C_n)$, with each $C_i$ a subset of $\{1,\ldots,n\}$, by having the $i$th car try to park in the leftmost available element of $C_i$. We further generalize this idea by restricting our sets to be of size $k$, intervals, and intervals of length $k$. In each of these cases we provide formulas for the number of such parking functions.

math.CO↗

arXiv2019

Slow Recurrences

For positive integers $α$ and $β$, we define an $(α,β)$-walk to be any sequence of positive integers satisfying $w_{k+2}=αw_{k+1}+βw_k$. We say that an $(α,β)$-walk is $n$-slow if $w_s=n$ with $s$ as large as possible. Slow $(1,1)$-walks have been investigated by several authors. In this paper we consider $(α,β)$-walks for arbitrary positive $α,β$. We derive a characterization theorem for these walks, and with this we prove several results concerning the total number of $n$-slow walks for a given $n$. In addition to this, we study the slowest $n$-slow walk for a given $n$ amongst all possible $α,β$.

math.NT↗

arXiv2019

Saturation Games for Odd Cycles

Given a family of graphs $\mathcal{F}$, we consider the $\mathcal{F}$-saturation game. In this game two players alternate adding edges to an initially empty graph on $n$ vertices, with the only constraint being that neither player can add an edge that creates a subgraph that lies in $\mathcal{F}$. The game ends when no more edges can be added to the graph. One of the players wishes to end the game as quickly as possible, while the other wishes to prolong the game. We let $sat_g(\mathcal{F};n)$ denote the number of edges that are in the final graph when both players play optimally. The $\{C_3\}$-saturation game was the first saturation game to be considered, but as of now the order of magnitude of $sat_g(\{C_3\},n)$ remains unknown. We consider a generalization of this game. Let $\mathcal{C}_{2k+1}:=\{C_3,\ C_5,\ldots,C_{2k+1}\}$. We prove that $sat_g(\mathcal{C}_{2k+1};n)\ge(\frac{1}{4}-ε_k)n^2+o(n^2)$ for all $k\ge 2$ and that $sat_g(\mathcal{C}_{2k+1};n)\le (\frac{1}{4}-ε'_k)n^2+o(n^2)$ for all $k\ge 4$, with $ε_k<\frac{1}{4}$ and $ε'_k>0$ constants tending to 0 as $k\to \infty$. In addition to this we prove $sat_g(\{C_{2k+1}\};n)\le \frac{4}{27}n^2+o(n^2)$ for all $k\ge 2$, and $sat_g(\mathcal{C}_\infty\setminus C_3;n)\le 2n-2$, where $\mathcal{C}_\infty$ denotes the set of all odd cycles.

math.CO↗

arXiv2019

Slow Fibonacci Walks

For a positive integer $n$, we study the number of steps to reach $n$ by a {\it Fibonacci walk} for some starting pair $a_1$ and $a_2$ satisfying the recurrence of $a_{k+2}=a_{k+1}+a_k$. The problem of slow Fibonacci walks, first suggested by Richard Stanley, is to determine the maximum number $s(n)$ of steps for such a Fibonacci walk ending at $n$. Stanley conjectured that for most $n$, there is a slow Fibonacci walk reaching $n = a_s$ with the property that $a_{s+1}$ is the integer closest to $ϕn$ where $ϕ=(1+\sqrt{5})/2$. We prove that this is true for only a positive fraction of $n$. We give explicit formulas for the choice of the starting pairs and the determination of $s(n)$ by giving a characterization theorem. We also derive a number of density results concerning the distribution of down and up cases (that is, those $n$ with $a_{s+1}=\lfloor ϕn\rfloor$ or $\lceil ϕn \rceil$, respectively), as well as for more general `paradoxical' cases.

math.NT↗

arXiv2019

Ballot Permutations and Odd Order Permutations

A permutation $π$ is ballot if, for all $k$, the word $π_1\cdots π_k$ has at least as many ascents as it has descents. Let $b(n)$ denote the number of ballot permutations of order $n$, and let $p(n)$ denote the number of permutations which have odd order in the symmetric group $S_n$. Callan conjectured that $b(n)=p(n)$ for all $n$, which was proved by Bernardi, Duplantier, and Nadeau. We propose a refinement of Callan's original conjecture. Let $b(n,d)$ denote the number of ballot permutations with $d$ descents. Let $p(n,d)$ denote the number of odd order permutations with $M(π)=d$, where $M(π)$ is a certain statistic related to the cyclic descents of $π$. We conjecture that $b(n,d)=p(n,d)$ for all $n$ and $d$. We prove this stronger conjecture for the cases $d=1,\ 2,\ 3$, and $d=\lfloor(n-1)/2\rfloor$, and in each of these cases we establish formulas for $b(n,d)$ involving Eulerian numbers and Eulerian-Catalan numbers.

math.CO↗

arXiv2019

Random graphs induced by Catalan pairs

We consider Catalan-pair graphs, a family of graphs that can be viewed as representing certain interactions between pairs of objects which are enumerated by the Catalan numbers. In this paper we study random Catalan-pair graphs and deduce various properties of these random graphs. In particular, we asymptotically determine the expected number of edges and isolated vertices, and more generally we determine the expected number of (induced) subgraphs isomorphic to a given connected graph.

math.CO↗

arXiv2017

Polynomial Relations Between Matrices of Graphs

We derive a correspondence between the eigenvalues of the adjacency matrix $A$ and the signless Laplacian matrix $Q$ of a graph $G$ when $G$ is $(d_1,d_2)$-biregular by using the relation $A^2=(Q-d_1I)(Q-d_2I)$. This motivates asking when it is possible to have $X^r=f(Y)$ for $f$ a polynomial, $r>0$, and $X,\ Y$ matrices associated to a graph $G$. It turns out that, essentially, this can only happen if $G$ is either regular or biregular.

math.CO↗

arXiv2017

Forbidden Families of Minimal Quadratic and Cubic Configurations

A matrix is \emph{simple} if it is a (0,1)-matrix and there are no repeated columns. Given a (0,1)-matrix $F$, we say a matrix $A$ has $F$ as a \emph{configuration}, denoted $F\prec A$, if there is a submatrix of $A$ which is a row and column permutation of $F$. Let $|A|$ denote the number of columns of $A$. Let $\mathcal{F}$ be a family of matrices. We define the extremal function $\text{forb}(m, \mathcal{F}) = \max\{|A|\colon A \text{ is an }m-\text{rowed simple matrix and has no configuration } F\in\mathcal{F}\}$. We consider pairs $\mathcal{F}=\{F_1,F_2\}$ such that $F_1$ and $F_2$ have no common extremal construction and derive that individually each $\text{forb}(m, F_i)$ has greater asymptotic growth than $\text{forb}(m, \mathcal{F})$, extending research started by Anstee and Koch.

math.CO↗