Related papers: The inducibility of oriented stars

Subdividing an edge $uv$ in a graph replaces it by a path $u w v$ with one new vertex. For a graph $H$, the \textsc{$H$-free Subdivision} problem asks whether, given a graph $G$ and an integer $k$, one can destroy all induced copies of $H$…

In new progress on conjectures of Stein, and Addario-Berry, Havet, Linhares Sales, Reed and Thomass\'e, we prove that every oriented graph with all in- and out-degrees greater than 5k/8 contains an alternating path of length k. This…

A set $C$ of unit vectors in $\mathbb{R}^d$ is called an $L$-spherical code if $x \cdot y \in L$ for any distinct $x,y$ in $C$. Spherical codes have been extensively studied since their introduction in the 1970's by Delsarte, Goethals and…

A typical Dirac-type problem in extremal graph theory is to determine the minimum degree threshold for a graph $G$ to have a spanning subgraph $H$, e.g. the Dirac theorem. A natural following up problem would be to seek an $H$-factor, which…

A $k$-star is a complete bipartite graph $K_{1,k}$. For a graph $G$, a $k$-star decomposition of $G$ is a set of $k$-stars in $G$ whose edge sets partition the edge set of $G$. If we weaken this condition to only demand that each edge of…

An oriented graph $\overrightarrow{G}$ is pushably $k$-critical if it is not pushably $k$-colorable, but every proper subgraph of $\overrightarrow{G}$ is. The main result of this article is that every pushably $3$-critical oriented graph on…

Let a 2 to 1 directed hypergraph be a 3-uniform hypergraph where every edge has two tail vertices and one head vertex. For any such directed hypergraph F let the nth extremal number of F be the maximum number of edges that any directed…

We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let $R$ and $B$ be…

We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2…

For a graph $G$ with the vertex set $V(G)$ and the edge set $E(G)$ and a star subgraph $S$ of $G$, let $\alpha_S(G)$ be the maximum number of vertices in $G$ such that no two of them are in the same star subgraph $S$ and $\theta_S(G)$ be…

For every $k \geq 3$, we determine the order of growth, up to polylogarithmic factors, of the number of orientations of the binomial random graph containing no directed cycle of length $k$. This solves a conjecture of Kohayakawa, Morris and…

We consider the Densest-Subgraph problem, where a graph and an integer k is given and we search for a subgraph on exactly k vertices that induces the maximum number of edges. We prove that this problem is NP-hard even when the input graph…

In this short note we consider the oriented vertex Tur\'an problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the…

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = \Pi_{i}^h a_i$ where…

Given an integer $\ell\ge 1$, a $(1,\le \ell)$-identifying code in a digraph is a dominating subset $C$ of vertices such that all distinct subsets of vertices of cardinality at most $\ell$ have distinct closed in-neighbourhood within $C$.…

The minimum positive $\ell$-degree $\delta^+_{\ell}(G)$ of a non-empty $k$-graph $G$ is the maximum $m$ such that every $\ell$-subset of $V(G)$ is contained in either none or at least $m$ edges of $G$; let $\delta^+_{\ell}(G):=0$ if $G$ has…

We study the ordered configuration spaces of star graphs. Inspired by the representation stability results of Church--Ellenberg--Farb for the ordered configuration space of a manifold and the edge stability results of…

Let $k\ge 3$ be an odd integer and let $n$ be a sufficiently large integer. We prove that the maximum number of edges in an $n$-vertex $k$-uniform hypergraph containing no $2$-regular subgraphs is $\binom{n-1}{k-1} + \lfloor\frac{n-1}{k}…

In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict…

In 2009, Kong, Wang, and Lee began work on the problem of finding the edge-balanced index sets of complete bipartite graphs $K_{m,n}$ by solving the cases where $n=1$, $2$, $3$, $4$, and $5$, and also the case where $m=n$. In an article…