Related papers: On polynomial degree-boundedness

Alon, Krivelevich, and Sudakov conjectured in 1999 that every $F$-free graph of maximum degree at most $\Delta$ has chromatic number $O(\Delta / \log \Delta)$. This was previously known only for almost bipartite graphs, that is, for…

In this paper, we show that for all $k\geq 10^8$, every graph with minimum degree $k$ and girth at least $10^8$ contains an induced subdivision of a $K_{k+1}$. This answers a problem asked by K\"uhn and Osthus (originally attributed to…

Let $H=(V,E)$ be an $s$-uniform hypergraph of order $n$ and $k\geq 0$ be an integer. A $k$-independent set $S\subseteq H$ is a set of vertices such that the maximum degree in the hypergraph induced by $S$ is at most $k$. Denoted by…

A graph $G$ contains another graph $H$ as an immersion if $H$ can be obtained from a subgraph of $G$ by splitting off edges and removing isolated vertices. There is an obvious necessary degree condition for the immersion containment: if $G$…

The domination polynomial of a graph $G$ is given by $D(G,x)=\sum_{k=0}^{n} d_k(G)x^k$ where $d_k(G)$ records the number of $k$-element dominating sets in $G$. A conjecture of Alikhani and Peng asserts that these polynomials have unimodal…

Let $G = (V, E)$ be a connected graph with maximum degree $k\geq 3$ distinct from $K_{k+1}$. Given integers $s \geq 2$ and $p_1,\ldots,p_s\geq 0$, $G$ is said to be $(p_1, \dots, p_s)$-partitionable if there exists a partition of $V$ into…

A class $\mathcal{F}$ of graphs is $\chi$-bounded if there is a function $f$ such that $\chi(H)\le f(\omega(H))$ for all induced subgraphs $H$ of a graph in $\mathcal{F}$. If $f$ can be chosen to be a polynomial, we say that $\mathcal{F}$…

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…

We show that for any constant $\Delta \ge 2$, there exists a graph $G$ with $O(n^{\Delta / 2})$ vertices which contains every $n$-vertex graph with maximum degree $\Delta$ as an induced subgraph. For odd $\Delta$ this significantly improves…

Generalising the Heilman-Lieb Theorem from statistical physics, Chudnovsky and Seymour [J. Combin. Theory Ser. B, 97(3):350--357] showed that the univariate independence polynomial of any claw-free graph has all of its zeros on the negative…

We initiate the systematic study of the following Tur\'an-type question. Suppose $\Gamma$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$…

Let $T$ be a forest. We study polynomially high-chromatic pure pairs in graphs with no $T$ as an induced subgraph ($T$-free graphs in other words), with applications to the polynomial Gy\'arf\'as-Sumner conjecture. In addition to reproving…

The conjecture of Bollob\'as and Koml\'os, recently proved by B\"ottcher, Schacht, and Taraz [Math. Ann. 343(1), 175--205, 2009], implies that for any $\gamma>0$, every balanced bipartite graph on $2n$ vertices with bounded degree and…

Given any integers $s,t\geq 2$, we show there exists some $c=c(s,t)>0$ such that any $K_{s,t}$-free graph with average degree $d$ contains a subdivision of a clique with at least $cd^{\frac{1}{2}\frac{s}{s-1}}$ vertices. In particular, when…

We prove that for every path H, and every integer d, there is a polynomial f such that every graph G with chromatic number greater than f(t) either contains H as an induced subgraph, or contains as a subgraph the complete d-partite graph…

A bull is a graph obtained from a four-vertex path by adding a vertex adjacent to the two middle vertices of the path. A graph $G$ is bull-free if no induced subgraph of $G$ is a bull. We prove that for all $k,t\in \mathbb{N}$, if $G$ is a…

We prove that if a graph contains the complete bipartite graph $K_{134, 12}$ as an induced minor, then it contains a cycle of length at most~12 or a theta as an induced subgraph. With a longer and more technical proof, we prove that if a…

A class of graphs is $\chi$-bounded if there is a function $f$ such that every graph $G$ in the class has chromatic number at most $f(\omega(G))$, where $\omega(G)$ is the clique number of $G$; the class is polynomially $\chi$-bounded if…

In this paper, we introduce a class of graphs which we call average hereditary graphs. Many graphs that occur in the usual graph theory applications belong to this class of graphs. Many popular types of graphs fall under this class, such as…

Let $P_k$ be a path, $C_k$ a cycle on $k$ vertices, and $K_{k,k}$ a complete bipartite graph with $k$ vertices on each side of the bipartition. We prove that (1) for any integers $k, t>0$ and a graph $H$ there are finitely many subgraph…