Induced subgraphs of graphs with large chromatic number. XII. Distant stars

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 families of trees, including trees of radius two, some special trees of radius three, and subdivided stars. These trees all have the property that their vertices of degree more than two are clustered quite closely together. In this paper, we prove the conjecture for two families of trees which do not have this restriction. As special cases, these families contain all double-ended brooms and two-legged caterpillars

A survey of χ\chi-boundedness

If a graph has bounded clique number, and sufficiently large chromatic number, what can we say about its induced subgraphs? Andr\'as Gy\'arf\'as made a number of challenging conjectures about this in the early 1980's, which have remained open until recently; but in the last few years there has been substantial progress. This is a survey of where we are now

Polynomial bounds for chromatic number. IV. A near-polynomial bound for excluding the five-vertex path

A graph G is H-free if it has no induced subgraph isomorphic to H. We prove that a $P_5$-free graph with clique number $\omega\ge 3$ has chromatic number at most $\omega^{\log_2(\omega)}$. The best previous result was an exponential upper bound $(5/27)3^{\omega}$, due to Esperet, Lemoine, Maffray, and Morel. A polynomial bound would imply that the celebrated Erdos-Hajnal conjecture holds for $P_5$, which is the smallest open case. Thus there is great interest in whether there is a polynomial bound for $P_5$-free graphs, and our result is an attempt to approach that

Polynomial χ\chi-binding functions for tt-broom-free graphs

For any positive integer $t$, a \emph{$t$-broom} is a graph obtained from $K_{1,t+1}$ by subdividing an edge once. In this paper, we show that, for graphs $G$ without induced $t$-brooms, we have $\chi(G) = o(\omega(G)^{t+1})$, where $\chi(G)$ and $\omega(G)$ are the chromatic number and clique number of $G$, respectively. When $t=2$, this answers a question of Schiermeyer and Randerath. Moreover, for $t=2$, we strengthen the bound on $\chi(G)$ to $7.5\omega(G)^2$, confirming a conjecture of Sivaraman. For $t\geq 3$ and \{$t$-broom, $K_{t,t}$\}-free graphs, we improve the bound to $o(\omega^{t-1+\frac{2}{t+1}})$.Comment: 14 pages, 1 figur

Induced subgraphs of graphs with large chromatic number. XIII. New brooms

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 function of their clique numbers. This remains open for general trees $T$, but has been proved for some particular trees. For $k\ge 1$, let us say a broom of length $k$ is a tree obtained from a $k$-edge path with ends $a,b$ by adding some number of leaves adjacent to $b$, and we call $a$ its handle. A tree obtained from brooms of lengths $k_1,...,k_n$ by identifying their handles is a $(k_1,...,k_n)$-multibroom. Kierstead and Penrice proved that every $(1,...,1)$-multibroom $T$ satisfies the Gy\'arf\'as-Sumner conjecture, and Kierstead and Zhu proved the same for $(2,...,2)$-multibrooms. In this paper give a common generalization: we prove that every $(1,...,1,2,...,2)$-multibroom satisfies the Gy\'arf\'as-Sumner conjecture

On coloring digraphs with forbidden induced subgraphs

This thesis mainly focuses on the structural properties of digraphs with high dichromatic number. The dichromatic number of a digraph $D$, denoted by \dichi(D), is designed to be the directed analog of the chromatic number of a graph $G$, denoted by $\chi(G)$. The field of $\chi$-boundedness studies the induced subgraphs that need to be present in a graph with high chromatic number. In this thesis, we study the equivalent of $\chi$-boundedness but with dichromatic number instead. In particular, we study the induced subgraphs of digraphs with high dichromatic number from two different perspectives which we describe below. First, we present results in the area of heroes. A digraph $H$ is a hero of a class of digraphs $\mathcal{C}$ if there exists a constant $c$ such that every $H$-free digraph $D\in \mathcal{C}$ has \dichi(D)\leq c. It is already known that when $\mathcal{C}$ is the family of $F$-free digraphs for some digraph $F$, the existence of heroes that are not transitive tournaments $TT_k$ implies that $F$ is the disjoint union of oriented stars. In this thesis, we narrow down the characterization of the digraphs $F$ which have heroes that are not transitive tournaments to the disjoint union of oriented stars of degree at most 4. Furthermore, we provide a big step towards the characterization of heroes in $\{rK_1+K_2 \}$-free digraphs, where $r\geq 1$. We achieve the latter by developing mathematical tools for proving that a hero in $F$-free digraphs is also a hero in $\{K_1+F\}$-free digraphs. Second, we present results in the area of \dichi-boundedness. In this area, we try to determine the classes of digraphs for which transitive tournaments are heroes. In particular, we ask whether, for a given class of digraphs $\mathcal{C}$, there exists a function $f$ such that, for every $k\geq 1$, \dichi(D)\leq f(k) whenever $D\in \mathcal{C}$ and $D$ is $TT_k$-free. We provide a comprehensive literature review of the subject and outline the $\chi$-boundedness results that have an equivalent result in \dichi-boundedness. We conclude by generalizing a key lemma in the literature and using it to prove $\{\mathcal{B}, \mathcal{B'} \}$-free digraphs are \dichi-bounded, where $\mathcal{B}$ and $\mathcal{B'}$ are small brooms whose orientations are related and have an additional particular property

Large structures in dense directed graphs

We answer questions in extremal combinatorics, for directed graphs. Specifically, we investigate which large tree-like directed graphs are contained in all dense directed graphs of large order. More precisely, let T be an oriented tree of order n; among others, we establish the following results. (1) We obtain a sufficient condition which ensures every tournament of order n contains T, and show that almost every tree possesses this property. (2) We prove that for all positive C, ɛ and sufficiently large n, every tournament of order (1+ɛ)n contains T if Δ(T)≤(log n)^C. (3) We prove that for all positive Δ, ɛ and sufficiently large n, every directed graph G of order n and minimum semidegree (1/2+ɛ)n contains T if Δ(T)≤Δ. (4) We obtain a sufficient condition which ensures that every directed graph G of order n with minimum semidegree at least (1/2+ɛ)n contains T, and show that almost every tree possesses this property. (5) We extend our method in (4) to a class of tree-like spanning graphs which includes all orientations of Hamilton cycles and large subdivisions of any graph. Result (1) confirms a conjecture of Bender and Wormald and settles a conjecture of Havet and Thomassé for almost every tree; (2) strengthens a result of Kühn, Mycroft and Osthus; (3) is a directed graph analogue of a classical result of Komlós, Sárközy and Szemerédi and is implied by (4) and (5) is of independent interest