arXiv subjects
Publications and source records attributed to Cory Palmer.
In this note we determine the maximum number of arcs in a digraph on $n$ vertices that does not contain a length-$k$ directed path $\overrightarrow{P}_{k+1}$ for $n \geq 50k^6$. This improves a theorem of Zhou and Li [$\textit{Graphs Combin.}$ 39(3), 2023] who proved the result with a superexponential threshold on $n$.
Let $\mathcal{H} \subseteq \binom{[n]}{r}$ be an $r$-uniform hypergraph on vertex set $[n] = \{1,2,\dots, n\}$. For an $r$-set of vertices $S \subseteq [n]$, the \emph{degree} of $S$ is defined as $\textrm{deg}(S)=\sum_{v \in S}\textrm{deg}(v)$ and the minimum of $\textrm{deg}(S)$ over all non-edge $r$-subsets $S \not \in E(\mathcal{H})$ of $V({\cal H})$ is the {\it Ore-degree} of ${\cal H}$, denoted by ${\sigma_r}({\cal H})$. We prove several Ore-degree results about existence of matchings in hypergraphs: (1) For $n\geq 2r+2$, if ${\cal H}$ is an intersecting $r$-uniform hypergraph on $n$ vertices, then $\sigma_r({\cal H})\leq r{n-2 \choose r-2}$, and there is equality only when ${\cal H}$ is a $1$-star. (2) For $r\geq 3$ and $n\geq 4r^2$, if is a non-trivial intersecting $r$-uniform hypergraph on $n$ vertices, then $\sigma_r({\cal H})\leq r\left({n-2 \choose r-2}-{n-r-2 \choose r-2}\right)$. (3) For $s\geq 2$ and $n\geq 3r^2(s-1)$, if ${\cal H}$ is an $r$-uniform hypergraph on $n$ vertices and $\sigma_r({\cal H})>r\left({n-1 \choose r-1}-{n-s \choose r-1}\right)$, then ${\cal H}$ contains $s$ pairwise disjoint edges.
For fixed graphs $H$ and $F$, the \emph{generalized Tur\'an number} $\mathrm{ex}(n,H,F)$ is the maximum possible number of copies of a subgraph $H$ in an $n$-vertex $F$-free graph. This article is a survey of this extremal function whose study was initiated in an influential 2016 article by Alon and Shikhelman (\emph{J. Combin. Theory, B}, {\bf 121}, 2016).
The minimum positive co-degree of a nonempty $r$-graph $H$, denoted by $\delta_{r-1}^+(H)$, is the largest integer $k$ such that for every $(r-1)$-set $S \subset V(H)$, if $S$ is contained in a hyperedge of $H$, then $S$ is contained in at least $k$ hyperedges of $H$. Given a family $\mathcal{F}$ of $r$-graphs, the positive co-degree Tur\'an function $\mathrm{co^+ex}(n,\mathcal{F})$ is the maximum of $\delta_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ containing no member of $\mathcal{F}$. The positive co-degree density of $\mathcal{F}$ is $\gamma^+(\mathcal{F}) = \underset{n \rightarrow \infty}{\lim} \frac{\mathrm{co^+ex}(n,\mathcal{F})}{n}.$ While the existence of $\gamma^+(\mathcal{F})$ is proved for all families $\mathcal{F}$, only few positive co-degree densities are known exactly. For a fixed $r \geq 2$, we call $\alpha \in [0,1]$ an achievable value if there exists a family of $r$-graphs $\mathcal{F}$ with $\gamma^+(\mathcal{F}) = \alpha$, and call $\alpha$ a jump if for some $\delta > 0$, there is no family $\mathcal{F}$ with $\gamma^+(\mathcal{F}) \in (\alpha, \alpha + \delta)$. Halfpap, Lemons, and Palmer showed that every $\alpha \in [0, \frac{1}{r})$ is a jump. We extend this result by showing that every $\alpha \in [0, \frac{2}{2r -1})$ is a jump. We also show that for $r = 3$, the set of achievable values is infinite, more precisely, $\frac{k-2}{2k-3}$ for every $k \geq 4$ is achievable. Finally, we determine two additional achievable values for $r=3$ using flag algebra calculations.
The Ramsey-Turán problem for $K_p$ asks for the maximum number of edges in an $n$-vertex $K_p$-free graph with independence number $o(n)$. In a natural generalization of the problem, cliques larger than the edge $K_2$ are counted. Let {\bf RT}$(n,\#K_q,K_p,o(n))$ denote the maximum number of copies of $K_q$ in an $n$-vertex $K_p$-free graph with independence number $o(n)$. Balogh, Liu and Sharifzadeh determined the asymptotics of {\bf RT}$(n,\# K_3,K_p,o(n))$. In this paper we will establish the asymptotics for counting copies of $K_4$, $K_5$, and for the case $p \geq 5q$. We also provide a family of counterexamples to a conjecture of Balogh, Liu and Sharifzadeh.
In a rainbow version of the classical Turán problem one considers multiple graphs on a common vertex set, thinking of each graph as edges in a distinct color, and wants to determine the minimum number of edges in each color which guarantees existence of a rainbow copy (having at most one edge from each graph) of a given graph. Here, we prove an optimal solution for this problem for any directed star and any number of colors.
The \emph{minimum positive co-degree} of a non-empty $r$-graph ${H}$, denoted $δ_{r-1}^+( {H})$, is the maximum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $ {H}$, then $S$ is contained in at least $k$ distinct hyperedges of $ {H}$. Given an $r$-graph ${F}$, we introduce the \emph{positive co-degree Turán number} $\mathrm{co^+ex}(n, {F})$ as the maximum positive co-degree $δ_{r-1}^+(H)$ over all $n$-vertex $r$-graphs $H$ that do not contain $F$ as a subhypergraph. In this paper we concentrate on the behavior of $\mathrm{co^+ex}(n, {F})$ for $3$-graphs $F$. In particular, we determine asymptotics and bounds for several well-known concrete $3$-graphs $F$ (e.g.\ $K_4^-$ and the Fano plane). We also show that, for $r$-graphs, the limit \[ γ^+(F) := \lim_{n \rightarrow \infty} \frac{\mathrm{co^+ex}(n, {F})}{n} \] exists, and ``jumps'' from $0$ to $1/r$, i.e., it never takes on values in the interval $(0,1/r)$. Moreover, we characterize which $r$-graphs $F$ have $γ^+(F)=0$. Our motivation comes primarily from the study of (ordinary) co-degree Turán numbers where a number of results have been proved that inspire our results.
Let $\mathcal{F}\subseteq \binom{[n]}{r}$ be an intersecting family of sets and let $Δ(\mathcal{F})$ be the maximum degree in $\mathcal{F}$, i.e., the maximum number of edges of $\mathcal{F}$ containing a fixed vertex. The \emph{diversity} of $\mathcal{F}$ is defined as $d(\mathcal{F}) := |\mathcal{F}| - Δ(\mathcal{F})$. Diversity can be viewed as a measure of distance from the `trivial' maximum-size intersecting family given by the Erd\H os-Ko-Rado Theorem. Indeed, the diversity of this family is $0$. Moreover, the diversity of the largest non-trivial intersecting family à la Hilton-Milner is $1$. It is known that the maximum possible diversity of an intersecting family $\mathcal{F}\subseteq \binom{[n]}{r}$ is $\binom{n-3}{r-2}$ as long as $n$ is large enough. We introduce a generalization called the \emph{$C$-weighted diversity} of $\mathcal{F}$ as $d_C(\mathcal{F}) := |\mathcal{F}| - C \cdot Δ(\mathcal{F})$. We determine the maximum value of $d_C(\mathcal{F})$ for intersecting families $\mathcal{F} \subseteq \binom{[n]}{r}$ and characterize the maximal families for $C\in \left[0,\frac{7}{3}\right)$ as well as give general bounds for all $C$. Our results imply, for large $n$, a recent conjecture of Frankl and Wang concerning a related diversity-like measure. Our primary technique is a variant of Frankl's Delta-system method.
In this note we examine the following random graph model: for an arbitrary graph $H$, with quadratic many edges, construct a graph $G$ by randomly adding $m$ edges to $H$ and randomly coloring the edges of $G$ with $r$ colors. We show that for $m$ a large enough constant and $r \geq 5$, every pair of vertices in $G$ are joined by a rainbow path, i.e., $G$ is {\it rainbow connected}, with high probability. This confirms a conjecture of Anastos and Frieze [{\it J. Graph Theory} {\bf 92} (2019)] who proved the statement for $r \geq 7$ and resolved the case when $r \leq 4$ and $m$ is a function of $n$.
We introduce, and partially resolve, a conjecture that brings a three-centuries-old derangements phenomenon and its much younger two-decades-old analogue under the same umbrella. Through a graph-theoretic lens, a derangement is a perfect matching in the complete bipartite graph $K_{n,n}$ with a disjoint perfect matching $M$ removed. Likewise, a deranged matching is a perfect matching in the complete graph $K_{2n}$ minus a perfect matching $M'$. With $\mathrm{pm}(\cdot)$ counting perfect matchings, the elder phenomenon takes the form $\mathrm{pm}(K_{n,n}-M)/\mathrm{pm}(K_{n,n})\to 1/e$ as $n\to\infty$ while its youthful analogue is $\mathrm{pm}(K_{2n}-M')/\mathrm{pm}(K_{2n})\to 1/\sqrt{e}$. These starting graphs are both $2n$-vertex `balanced complete $r$-partite' graphs $K_{r \times {2n}/{r}}$, respectively with $r=2$ and $r=2n$. We conjecture that $\mathrm{pm}(K_{r\times{2n}/r}-M)/\mathrm{pm}(K_{r\times{2n}/r})\sim e^{-r/(2r-2)}$ as $n\to\infty$ and establish several substantive special cases thereof. For just two examples, $r=3$ yields the limit $e^{-3/4}$ while $r=n$ results again in $e^{-1/2}$. Our tools blend combinatorics and analysis in a medley incorporating Inclusion-Exclusion and Tannery's Theorem.
We connect two classical results in extremal graph theory concerning the number of maximal independent sets. The maximum number mis$(n)$ of maximal independent sets in an $n$-vertex graph was determined by Moon and Moser. The maximum number mis$_\bigtriangleup(n)$ of maximal independent sets in an $n$-vertex triangle-free graph was determined by Hujter and Tuza. We determine the maximum number mis$_t(n)$ of maximal independent sets in an $n$-vertex graph containing no induced triangle matching of size $t+1$. We also reprove a stability result of Kahn and Park on the maximum number mis$_{\bigtriangleup,t}(n)$ of maximal independent sets in an $n$-vertex triangle-free graphs containing no induced matching of size $t+1$.
We improve the upper bound for the maximum possible number of stable matchings among $n$ jobs and $n$ applicants from $131072^n+O(1)$ to $3.55^n+O(1)$. To establish this bound, we state a novel formulation of a certain entropy bound that is easy to apply and may be of independent interest in counting other combinatorial objects
Let $\mathcal{H}$ be an $r$-uniform hypergraph. The \emph{minimum positive co-degree} of $\mathcal{H}$, denoted by $δ_{r-1}^+(\mathcal{H})$, is the minimum $k$ such that if $S$ is an $(r-1)$-set contained in a hyperedge of $\mathcal{H}$, then $S$ is contained in at least $k$ hyperedges of $\mathcal{H}$. For $r\geq k$ fixed and $n$ sufficiently large, we determine the maximum possible size of an intersecting $r$-uniform $n$-vertex hypergraph with minimum positive co-degree $δ_{r-1}^+(\mathcal{H}) \geq k$ and characterize the unique hypergraph attaining this maximum. This generalizes the Erd\H os-Ko-Rado theorem which corresponds to the case $k=1$. Our proof is based on the delta-system method.
An edge-colored graph $F$ is {\it rainbow} if each edge of $F$ has a unique color. The {\it rainbow Turán number} $\mathrm{ex}^*(n,F)$ of a graph $F$ is the maximum possible number of edges in a properly edge-colored $n$-vertex graph with no rainbow copy of $F$. The study of rainbow Turán numbers was introduced by Keevash, Mubayi, Sudakov, and Verstraëte. Johnson and Rombach introduced the following rainbow-version of generalized Turán problems: for fixed graphs $H$ and $F$, let $\mathrm{ex}^*(n,H,F)$ denote the maximum number of rainbow copies of $H$ in an $n$-vertex properly edge-colored graph with no rainbow copy of $F$. In this paper we investigate the case $\mathrm{ex}^*(n,C_\ell,P_\ell)$ and give a general upper bound as well as exact results for $\ell = 3,4,5$. Along the way we establish a new best upper bound on $\mathrm{ex}^*(n,P_5)$. Our main motivation comes from an attempt to improve bounds on $\mathrm{ex}^*(n,P_\ell)$, which has been the subject of several recent manuscripts.
Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Turán graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large enough). We say that such a graph $H$ is $F$-Turán-good. In addition to some general results, we give (among others) the following concrete results: (i) For every complete multipartite graph $H$, there is $k$ large enough such that $H$ is $K_k$-Turán-good. (ii) The path $P_3$ is $F$-Turán-good for $F$ with $χ(F) \geq 4$. (iii) The path $P_4$ and cycle $C_4$ are $C_5$-Turán-good. (iv) The cycle $C_4$ is $F_2$-Turán-good where $F_2$ is the graph of two triangles sharing exactly one vertex.
Fix a graph $F$. We say that a graph is {\it $F$-free} if it does not contain $F$ as a subgraph. The {\it Turán number} of $F$, denoted $\mathrm{ex}(n,F)$, is the maximum number of edges possible in an $n$-vertex $F$-free graph. The study of Turán numbers is a central problem in graph theory. The goal of this paper is to generalize a theorem of Lidický, Liu and Palmer [{\it Electron.\ J.\ of Combin.}\ {\bf 20} (2016)] that determines $\mathrm{ex}(n,F)$ for $F$ a forest of stars. In particular, we consider generalizations of the problem to three different well-studied hypergraph settings and in each case we prove an asymptotic result for all reasonable parameters defining our "star forests".
Alon and Shikhelman initiated the systematic study of the following generalized Turán problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Turán number of $F$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $F$. The study of rainbow Turán problems was initiated by Keevash, Mubayi, Sudakov and Verstraëte. Motivated by the above problems, we study the following problem: What is the maximum number of copies of $F$ in a properly edge-colored graph on $n$ vertices without a rainbow copy of $F$? We establish several results, including when $F$ is a path, cycle or tree.
Fix graphs $F$ and $H$. Let $\mathrm{ex}(n,H,F)$ denote the maximum number of copies of a graph $H$ in an $n$-vertex $F$-free graph. In this note we will give a new general supersaturation result for $\mathrm{ex}(n,H,F)$ in the case when $χ(H) < χ(F)$ as well as a new proof of a stability theorem for $\mathrm{ex}(n,K_r,F)$.