Related papers: Induced subdivisions of $K_{d+1}$ in graphs of hig…

In this paper, we prove that there exists an absolute constant $g_0$ such that, for every integer $k\ge 3$, every graph $G$ with $\delta(G)\ge k$ and $g(G)\ge g_0$ contains an induced subdivision of $K_{k+1}$. This answers, in a strong…

We prove that for every graph H and for every integer s, the class of graphs that do not contain K_s, K_{s,s}, or any subdivision of H as an induced subgraph has bounded expansion; this strengthens a result of Kuhn and Osthus. The argument…

A graph $H$ is an induced minor of a graph $G$ if $H$ can be obtained from $G$ by vertex deletions and edge contractions. We show that there is a function $f(k, d) = O(k^{10} + 2^{d^5})$ so that if a graph has treewidth at least $f(k, d)$…

Erd\H{o}s, Fajtlowicz and Staton asked for the least integer $f(k)$ such that every graph with more than $f(k)$ vertices has an induced regular subgraph with at least $k$ vertices. Here we consider the following relaxed notions. Let $g(k)$…

We prove that every bipartite graph of sufficiently large average degree has either a $K_{t,t}$-subgraph or an induced subgraph of average degree at least $t$ and girth at least $6$. We conjecture that "$6$" can be replaced by "$k$", which…

We prove that every graph of minimum degree at least $d \ge 1$ contains a subdivision of some maximal 3-degenerate graph of order $d+1$. This generalizes the classic results of Dirac ($d=3$) and Pelik\'an ($d=4$). We conjecture that for any…

We prove that every graph $G$ on $n$ vertices with no isolated vertices contains an induced subgraph of size at least $n/10000$ with all degrees odd. This solves an old and well-known conjecture in graph theory.

In this paper we prove that for every $s\geq 2$ and every graph $H$ the following holds. Let $G$ be a graph with average degree $\Omega_H(s^{C|H|^2})$, for some absolute constant $C>0$, then $G$ either contains a $K_{s,s}$ or an induced…

In this note we prove that for every integer $k$, there exist constants $g_{1}(k)$ and $g_{2}(k)$ such that the following holds. If $G$ is a graph on $n$ vertices with maximum degree $\Delta$ then it contains an induced subgraph $H$ on at…

We present exact and heuristic algorithms that find, for a given family of graphs, a graph that contains each member of the family as an induced subgraph. For $0 \leq k \leq 6$, we give the minimum number of vertices $f(k)$ in a graph…

Let $n\geq 6,k\geq 0$ be two integers. Let $H$ be a graph of order $n$ with $k$ components, each of which is an even cycle of length at least $6$ and $G$ be a bipartite graph with bipartition $(X,Y)$ such that $|X|=|Y|\geq n/2$. In this…

Ramsey proved that for every positive integer $n$, every sufficiently large graph contains an induced $K_n$ or $\overline{K}_n$. Among the many extensions of Ramsey's Theorem there is an analogue for connected graphs: for every positive…

A set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh and Staden proved that among all graphs with minimum degree $d$, $K_{d+1}$ minimises the number of…

Kelly, Kuehn and Osthus conjectured that for any l>3 and the smallest number k>2 that does not divide l, any large enough oriented graph G with minimum indegree and minimum outdegree at least \lfloor |V(G)|/k\rfloor +1 contains a directed…

We prove that for all $0\leq t\leq k$ and $d\geq 2k$, every graph $G$ with treewidth at most $k$ has a `large' induced subgraph $H$, where $H$ has treewidth at most $t$ and every vertex in $H$ has degree at most $d$ in $G$. The order of $H$…

The girth of a graph is the length of its shortest cycle. Due to its relevance in graph theory, network analysis and practical fields such as distributed computing, girth-related problems have been object of attention in both past and…

In this paper we show that there exists a constant $C>0$ such that for any graph $G$ on $Ck\ln k$ vertices either $G$ or its complement $\bar{G}$ has an induced subgraph on $k$ vertices with minimum degree at least $\frac12(k-1)$. This…

For every graph $G$, let $\alpha(G)$ denote its independence number. What is the minimum of the maximum degree of an induced subgraph of $G$ with $\alpha(G)+1$ vertices? We study this question for the $n$-dimensional Hamming graph over an…

We show that for any positive integers $g$ and $t$, there is a $K_{6}^{(1)}$-induced-minor-free graph of girth at least $g$ that is not a region intersection graph over the class of $K_t$-minor-free graphs. This answers in a strong form the…

Koml\'os conjectured in 1981 that among all graphs with minimum degree at least $d$, the complete graph $K_{d+1}$ minimises the number of Hamiltonian subsets, where a subset of vertices is Hamiltonian if it contains a spanning cycle. We…