Related papers: A survey of $\chi$-boundedness
A class of graphs is $\chi$-bounded if there is a function $f$ such that $\chi(G)\le f(\omega(G))$ for every induced subgraph $G$ of every graph in the class, where $\chi,\omega$ denote the chromatic number and clique number of $G$…
Suppose a graph has no large balanced bicliques, but has large minimum degree. Then what can we say about its induced subgraphs? This question motivates the study of degree-boundedness, which is like $\chi$-boundedness but for minimum…
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 $\ell$.
Resolving a 1985 open problem of Gy\'arf\'as, we prove that chromatic number is polynomially bounded by clique number for graphs with no induced five-vertex path $P_5$. Our approach introduces a chromatic density framework involving…
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…
We study graphs whose chromatic number is close to the order of the graph (the number of vertices). Both when the chromatic number is a constant multiple of the order and when the difference of the chromatic number and the order is a small…
Gyarfas conjectured in 1985 that for all $k$, $l$, every graph with no clique of size more than $k$ and no odd hole of length more than $l$ has chromatic number bounded by a function of $k$ and $l$. We prove three weaker statements: (1)…
Here we prove that a graph without some three induced subgraphs has chromatic number at the most equal to its maximum clique size plus one. Further we show that the bounds are tight and give examples to show that each of the three forbidden…
We prove a conjecture of Andras Gyarfas, that for all k,t, every graph with clique number at most k and sufficiently large chromatic number has an odd hole of length at least t.
Although the chromatic number of a graph is not known in general, attempts have been made to find good bounds for the number. Here we prove that for a graph G with two forbidden subgraphs and maximum degree less than or equal to 2{\omega} -…
A graph class is $\chi$-bounded if the only way to force large chromatic number in graphs from the class is by forming a large clique. In the 1970s, Erd\H{o}s conjectured that intersection graphs of straight-line segments in the plane are…
We discuss some problems related to induced subgraphs. The first problem is about getting a good upper bound for the chromatic number in terms of the clique number for graphs in which every induced cycle has length $3$ or $4$. The second…
Gy\'arf\'as and Sumner independently conjectured that for every tree $T$, the class of graphs not containing $T$ as an induced subgraph is $\chi$-bounded, that is, the chromatic numbers of graphs in this class are bounded above by a…
Erd\H{o}s proved that there are graphs with arbitrarily large girth and chromatic number. We study the extension of this for generalized chromatic numbers.
We prove that, for every graph $F$ with at least one edge, there is a constant $c_F$ such that there are graphs of arbitrarily large chromatic number and the same clique number as $F$ in which every $F$-free induced subgraph has chromatic…
The Gyarfas-Sumner conjecture asserts that if H is a tree then every graph with bounded clique number and very large chromatic number contains H as an induced subgraph. This is still open, although it has been proved for a few simple…
A class of graphs G is chi-bounded if the chromatic number of graphs in G is bounded by a function of the clique number. We show that if a class G is chi-bounded,then every class of graphs admitting a decomposition along cuts of small rank…
In the way of proving Kneser's conjecture, L\'{a}szl\'{o} Lov\'{a}sz settled out a new lower bound for the chromatic number. He showed that if neighborhood complex $\mathcal{N}(G)$ of a graph $G$ is topologically $k$-connected, then its…
A variety of powerful extremal results have been shown for the chromatic number of triangle-free graphs. Three noteworthy bounds are in terms of the number of vertices, edges, and maximum degree given by Poljak \& Tuza (1994), and…
Fix an oriented graph H, and let G be a graph with bounded clique number and very large chromatic number. If we somehow orient its edges, must there be an induced subdigraph isomorphic to H? Kierstead and Rodl raised this question for two…