Some results on concatenating bipartite graphs

We consider two functions $\phi$ and $\psi$, defined as follows. Let $x,y \in (0,1]$ and let $A,B,C$ be disjoint nonempty subsets of a graph $G$, where every vertex in $A$ has at least $x|B|$ neighbors in $B$, and every vertex in $B$ has at least $y|C|$ neighbors in $C$. We denote by $\phi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v \in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. If in addition we require that every vertex in $B$ has at least $x|A|$ neighbors in $A$, and every vertex in $C$ has at least $y|B|$ neighbors in $C$, we denote by $\psi(x,y)$ the maximum $z$ such that, in all such graphs $G$, there is a vertex $v \in C$ that is joined to at least $z|A|$ vertices in $A$ by two-edge paths. In their recent paper, M. Chudnovsky, P. Hompe, A. Scott, P. Seymour, and S. Spirkl introduced these functions, proved some general results about them, and analyzed when they are greater than or equal to $1/2, 2/3,$ and $1/3$. Here, we extend their results by analyzing when they are greater than or equal to $3/4, 2/5,$ and $3/5$.Comment: arXiv admin note: text overlap with arXiv:1902.10878 by other author

Proof of the Caccetta-Haggkvist conjecture for digraphs with small independence number

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge n/k$ for all $v \in V(G)$, then G contains a directed cycle of length at most $k$. In [2], N. Lichiardopol proved that this conjecture is true for digraphs with independence number equal to two. In this paper, we generalize that result, proving that the conjecture is true for digraphs with independence number at most $(k+1)/2$

Aharoni's rainbow cycle conjecture holds up to an additive constant

In 2017, Aharoni proposed the following generalization of the Caccetta-H\"{a}ggkvist conjecture: if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $\lceil n/r \rceil$. In this paper, we prove that, for fixed $r$, Aharoni's conjecture holds up to an additive constant. Specifically, we show that for each fixed $r \geq 1$, there exists a constant $c_r$ such that if $G$ is a simple $n$-vertex edge-colored graph with $n$ color classes of size at least $r$, then $G$ contains a rainbow cycle of length at most $n/r + c_r$.Comment: 10 pages, 0 figures. Upgraded the main result from the previous version so that it now holds up to an additive constan

Further approximations for Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. With Pelik\'anov\'a and Pokorn\'a, we showed that this conjecture is true if each color class has size ${\Omega}(k\log k)$. In this paper, we present a proof of the conjecture if each color class has size ${\Omega}(k)$, which improved the previous result and is only a constant factor away from Aharoni's conjecture. We also consider what happens when the condition on the number of colors is relaxed

Cycles and coloring in graphs and digraphs

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

Improved bounds for the triangle case of Aharoni's rainbow generalization of the Caccetta-H\"{a}ggkvist conjecture

For a digraph $G$ and $v \in V(G)$, let $\delta^+(v)$ be the number of out-neighbors of $v$ in $G$. The Caccetta-H\"{a}ggkvist conjecture states that for all $k \ge 1$, if $G$ is a digraph with $n = |V(G)|$ such that $\delta^+(v) \ge k$ for all $v \in V(G)$, then $G$ contains a directed cycle of length at most $\lceil n/k \rceil$. Aharoni proposed a generalization of this conjecture, that a simple edge-colored graph on $n$ vertices with $n$ color classes, each of size at least $k$, has a rainbow cycle of length at most $\lceil n/k \rceil$. Let us call $(\alpha, \beta)$ \emph{triangular} if every simple edge-colored graph on $n$ vertices with at least $\alpha n$ color classes, each with at least $\beta n$ edges, has a rainbow triangle. Aharoni, Holzman, and DeVos showed the following: $(9/8,1/3)$ is triangular; $(1,2/5)$ is triangular. In this paper, we improve those bounds, showing the following: $(1.1077,1/3)$ is triangular; $(1,0.3988)$ is triangular. Our methods give results for infinitely many pairs $(\alpha, \beta)$, including $\beta < 1/3$; we show that $(1.3481,1/4)$ is triangular.Comment: Accepted manuscript; see DOI for journal versio

A counterexample to a conjecture about triangle-free induced subgraphs of graphs with large chromatic number

We prove that for every $n$, there is a graph $G$ with $\chi(G) \geq n$ and $\omega(G) \leq 3$ such that every induced subgraph $H$ of $G$ with $\omega(H) \leq 2$ satisfies $\chi(H) \leq 4$. This disproves a well-known conjecture. Our construction is a digraph with bounded clique number, large dichromatic number, and no induced directed cycles of odd length at least 5.Comment: Moving one of the results to a different paper, where it fits bette

Digraphs with All Induced Directed Cycles of the Same Length are not → χ -Bounded

For t > 2, let us call a digraph D t-chordal if all induced directed cycles in D have length equal to t. In an earlier paper, we asked for which t it is true that t-chordal graphs with bounded clique number have bounded dichromatic number. Recently, Aboulker, Bousquet, and de Verclos answered this in the negative for t = 3, that is, they gave a construction of 3-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number. In this paper, we extend their result, giving for each t > 3 a construction of t-chordal digraphs with clique number at most 3 and arbitrarily large dichromatic number, thus answering our question in the negative. On the other hand, we show that a more restricted class, digraphs with no induced directed cycle of length less than t, and no induced directed t-vertex path, have bounded dichromatic number if their clique number is bounded. We also show the following complexity result: for fixed t > 2, the problem of determining whether a digraph is t-chordal is coNP-complete.This research was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), RGPIN-2020-03912