Cycles and coloring in graphs and digraphs

Abstract

We show results in areas related to extremal problems in directed graphs. The first concerns a rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture, made by Aharoni. The Caccetta-H\"{a}ggkvist conjecture states that if $G$ is a simple digraph on $n$ vertices with minimum out-degree at least $k$, then there exists a directed cycle in $G$ of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this well-known conjecture, namely that if $G$ is a simple edge-colored graph (not necessarily properly colored) on $n$ vertices with $n$ color classes each of size at least $k$, then there exists a rainbow cycle in $G$ of length at most $\lceil n/k \rceil$. In this thesis, we first prove that if $G$ is an edge-colored graph on $n$ vertices with $n$ color classes each of size at least $\Omega(k \log{k})$, then $G$ has a rainbow cycle of length at most $\lceil n/k \rceil$. Then, we develop more techniques to prove the stronger result that if there are $n$ color classes, each of size at least $\Omega(k)$, then there is a rainbow cycle of length at most $\lceil n/k \rceil$. Finally, we improve upon existing bounds for the triangle case, showing that if there are $n$ color classes of size at least $0.3988n$, then there exists a rainbow triangle, and also if there are $1.1077n$ color classes of size at least $n/3$, then there is a rainbow triangle.

Let $\chi(G)$ denote the \emph{chromatic number} of a graph $G$ and let $\omega(G)$ denote the \emph{clique number}. Similarly, let $\dichi(D)$ denote the \emph{dichromatic number} of a digraph $D$ and let $\omega(D)$ denote the clique number of the underlying undirected graph of $D$. In the second part of this thesis, we consider questions of $\chi$-boundedness and $\dichi$-boundedness. In the undirected setting, the question of $\chi$-boundedness concerns, for a class $\mathcal{C}$ of graphs, for what functions $f$ (if any) is it true that $\chi(G) \le f(\omega(G))$ for all graphs $G \in \mathcal{C}$. In a similar way, the notion of $\dichi$-boundedness refers to, given a class $\mathcal{C}$ of digraphs, for what functions $f$ (if any) is it true that $\dichi(D) \le f(\omega(D))$ for all digraphs $D \in \mathcal{C}$. It was a well-known conjecture, sometimes attributed to Esperet, that for all $k,r \in \mathbb{N}$ there exists $n$ such that in every graph with $G$ with $\chi(G) \ge n$ and $\omega(G) \le k$, there exists an induced subgraph $H$ of $G$ with $\chi(H) \ge r$ and $\omega(H) = 2$. We disprove this conjecture. Then, we examine the class of $k$-chordal digraphs, which are digraphs such that all induced directed cycles have length equal to $k$. We show that for $k \ge 3$, the class of $k$-chordal digraphs is not $\dichi$-bounded, generalizing a result of Aboulker, Bousquet, and de Verclos in [1] for $k=3$. Then we give a hardness result for determining whether a digraph is $k$-chordal, and finally we show a result in the positive direction, namely that the class of digraphs which are $k$-chordal and also do not contain an induced directed path on $k$ vertices is $\dichi$-bounded.

We discuss the work of others stemming from and related to our results in both areas, and outline directions for further work.