Related papers: On vertex sets inducing tangles

Erd\H{o}s and Hajnal conjectured that for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or a stable set of size at least $|G|^c$ (a graph is $H$-free if it has no induced subgraph isomorphic to $H$).…

The Induced Graph Matching problem asks to find k disjoint induced subgraphs isomorphic to a given graph H in a given graph G such that there are no edges between vertices of different subgraphs. This problem generalizes the classical…

A subset $S$ of a vertex set of a graph $G$ is a total $(k,r)$-dominating set if every vertex $u \in V(G)$ is within distance $k$ of at least $r$ vertices in $S$. The minimum cardinality among all total $(k,r)$-dominating sets of $G$ is…

In connection with his solution of the Sensitivity Conjecture, Hao Huang (arXiv: 1907.00847, 2019) asked the following question: Given a graph $G$ with high symmetry, what can we say about the smallest maximum degree of induced subgraphs of…

Bhawalkar, Kleinberg, Lewi, Roughgarden, and Sharma [ICALP 2012] introduced the Anchored k-Core problem, where the task is for a given graph G and integers b, k, and p to find an induced subgraph H with at least p vertices (the core) such…

For any graph (hypergraph) $G$ with vertex set $V$ and edge set $E$, we define its incidence bipartite graph $\mathcal{I}(G)$ as the bipartite graph with bipartition $(E, V)$, where an edge $e \in E$ is adjacent to a vertex $v \in V$ in…

A graph is called $(k,t)$-regular if it is $k$-regular and the induced subgraph on the neighbourhood of every vertex is $t$-regular. We find new conditions on $(k,t)$ for the existence of such graphs and provide a wide range of examples.

In a graph story the vertices enter a graph one at a time and each vertex persists in the graph for a fixed amount of time $\omega$, called viewing window. At any time, the user can see only the drawing of the graph induced by the vertices…

A graph $G=(V,E)$ is called $(k,k')$-choosable if for any total list assignment $L$ which assigns to each vertex $v$ a set $L(v)$ of $k$ real numbers, and assigns to each edge $e$ a set $L(e)$ of $k'$ real numbers, there is a mapping…

We prove a 1985 conjecture of Gy\'arf\'as that for all $k,\ell$, every graph with sufficiently large chromatic number contains either a complete subgraph with $k$ vertices or an induced cycle of length at least $\ell$.

Let $G$ be an infinite graph whose vertex set is the set of positive integers, and let $G_n$ be the subgraph of $G$ induced by the vertices $\{1,2, \dots , n \}$. An increasing path of length $k$ in $G$, denoted $I_k$, is a sequence of…

Let $G=(V,E)$ be a simple undirected graph. The open neighbourhood of a vertex $v$ in $G$ is defined as $N_G(v)=\{u\in V~|~ uv\in E\}$; whereas the closed neighbourhood is defined as $N_G[v]= N_G(v)\cup \{v\}$. For an integer $k$, a subset…

A seminal result by Koml\'os, Sark\"ozy, and Szemer\'edi states that if a graph $G$ with $n$ vertices has minimum degree at least $kn/(k + 1)$, for some $k \in \mathbb{N}$ and $n$ sufficiently large, then it contains the $k$-th power of a…

Given a graph $G$, the $k$-dominating graph of $G$, $D_k(G)$, is defined to be the graph whose vertices correspond to the dominating sets of $G$ that have cardinality at most $k$. Two vertices in $D_k(G)$ are adjacent if and only if the…

Let $H$ be a fixed graph whose edges are colored red and blue and let $\beta \in [0,1]$. Let $I(H, \beta)$ be the (asymptotically normalized) maximum number of copies of $H$ in a large red/blue edge-colored complete graph $G$, where the…

We consider the problem of covering a graph with a given number of induced subgraphs so that the maximum number of vertices in each subgraph is minimized. We prove NP-completeness of the problem, prove lower bounds, and give approximation…

A famous conjecture of Gy\'arf\'as and Sumner states for any tree $T$ and integer $k$, if the chromatic number of a graph is large enough, either the graph contains a clique of size $k$ or it contains $T$ as an induced subgraph. We discuss…

The $(\kappa,\ell)$-edge-inducibility problem asks for the maximum number of $\kappa$-subsets inducing exactly $\ell$ edges that a graph of given order $n$ can have. Using flag algebras and stability approach, we resolve this problem for…

We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals…

Tangles, as introduced by Robertson and Seymour, were designed as an indirect way of capturing clusters in graphs and matroids. They have since been shown to capture clusters in much broader discrete structures too. But not all tangles are…