Related papers: Exact Counts of $C_{4}$s in Blow-Up Graphs

The \emph{blow-up} of a graph $H$ is the graph obtained from replacing each edge in $H$ by a clique of the same size where the new vertices of the cliques are all different. Erd\H{o}s et al. and Chen et al. determined the extremal number of…

Let $C(n)$ denote the maximum number of induced copies of 5-cycles in graphs on $n$ vertices. For $n$ large enough, we show that $C(n)=a\cdot b\cdot c \cdot d \cdot e + C(a)+C(b)+C(c)+C(d)+C(e)$, where $a+b+c+d+e = n$ and $a,b,c,d,e$ are as…

In a recent work, Allen, B\"{o}ttcher, H\`{a}n, Kohayakawa, and Person provided a first general analogue of the blow-up lemma applicable to sparse (pseudo)random graphs thus generalising the classic tool of Koml\'{o}s, S\'{a}rk\"{o}zy, and…

The blow-up of a graph is obtained by replacing every vertex with a finite collection of copies so that the copies of two vertices are adjacent if and only if the originals are. If every vertex is replaced with the same number of copies,…

For a planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $\operatorname{\mathbf{N}}_{\mathcal P}(n,P_7)\sim{4\over…

We propose an improved algorithm for counting the number of Hamiltonian cycles in a directed graph. The basic idea of the method is sequential acceptance/rejection, which is successfully used in approximating the number of perfect matchings…

Jacob Fox, C. Seshadhri, Tim Roughgarden, Fan Wei, and Nicole Wein introduced the model of $c$-closed graphs--a distribution-free model motivated by triadic closure, one of the most pervasive structural signatures of social networks. While…

Blow-up in graph theory is a procedure in which each vertex is replaced by copies of itself, and two copies are adjacent if and only if the original vertices are adjacent. In this paper, we extend the concept of graph blow-up to a more…

We show that every graph $G$ on $n$ vertices with $\delta(G) \geq (1/2+\varepsilon)n$ is spanned by a complete blow-up of a cycle with clusters of nearly uniform size $\Omega(\log n)$. The proof is based on a recently introduced approach…

For large $n$ we determine the maximum number of induced 6-cycles which can be contained in a planar graph on $n$ vertices, and we classify the graphs which achieve this maximum. In particular we show that the maximum is achieved by the…

Let $H$ be a graph and $p$ be an integer. The edge blow-up $H^p$ of $H$ is the graph obtained from replacing each edge in $H$ by a copy of $K_p$ where the new vertices of the cliques are all distinct. Let $C_k$ and $P_k$ denote the cycle…

We study four-cycle counting in arbitrary order graph streams. We present a 3-pass algorithm for $(1+\varepsilon)$-approximating the number of four-cycles using $\widetilde{O}(m/\sqrt{T})$ space, where $m$ is the number of edges and $T$ the…

In 1984, Erd\H{o}s conjectured that the number of pentagons in any triangle-free graph on $n$ vertices is at most $(n/5)^5$, which is sharp by the balanced blow-up of a pentagon. This was proved by Grzesik, and independently by Hatami,…

We consider the problem of counting 4-cycles ($C_4$) in an undirected graph $G$ of $n$ vertices and $m$ edges (in bipartite graphs, 4-cycles are also often referred to as $\textit{butterflies}$). Most recently, Wang et al. (2019, 2022)…

A cornerstone of extremal graph theory due to Erd\H{o}s and Stone states that the edge density which guarantees a fixed graph $F$ as subgraph also asymptotically guarantees a blow-up of $F$ as subgraph. It is natural to ask whether this…

A graph $G=(V,E)$ is said to be a \textit{$k$-threshold graph} with \textit{thresholds} $\theta_1<\theta_2<...<\theta_k$ if there is a map $r: V \longrightarrow \mathbb{R}$ such that $uv\in E$ if and only if $\theta_i\le r(u)+r(v)$ holds…

A {\em hole} is an induced cycle of length at least 4, an {\em even hole} is a hole of even length, and a {\em cap} is a graph obtained from a hole by adding an additional vertex which is adjacent exactly to two adjacent vertices of the…

A constructive method is provided that outputs a directed graph which is named a broken crown graph, containing $5n-9$ vertices and $k$ Hamiltonian cycles for any choice of integers $n \geq k \geq 4$. The construction is not designed to be…

A graph construction that produces a k-regular graph on n vertices for any choice of k >= 3 and n = m(k+1) for integer m >= 2 is described. The number of Hamiltonian cycles in such graphs can be explicitly determined as a function of n and…

In this paper, we present examples of the cyclic sieving phenomenon coming from studying independent sets in graphs of a fixed size k. Given a graph G, and a cyclic group C acting on the graph, then C also acts on the collection of…