On coloring digraphs with forbidden induced subgraphs

Abstract

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.