Tag: LindaCook

An oriented graph is a digraph that does not contain a directed cycle of length two. An (oriented) graph $D$ is $H$-free if $D$ does not contain $H$ as an induced sub(di)graph.

The Gyárfás-Sumner conjecture is a widely-open conjecture on simple graphs, which states that for any forest $F$, there is some function $f$ such that every $F$-free graph $G$ with clique number $\omega(G)$ has chromatic number at most $f(\omega(G))$.

Aboulker, Charbit, and Naserasr [Extension of Gyárfás-Sumner Conjecture to Digraphs; E-JC 2021] proposed an analogue of this conjecture to the dichromatic number of oriented graphs. The dichromatic number of a digraph $D$ is the minimum number of colors required to color the vertex set of $D$ so that no directed cycle in $D$ is monochromatic. Aboulker, Charbit, and Naserasr’s $\overrightarrow{\chi}$-boundedness conjecture states that for every oriented forest $F$, there is some function $f$ such that every $F$-free oriented graph $D$ has dichromatic number at most $f(\omega(D))$, where $\omega(D)$ is the size of a maximum clique in the graph underlying $D$.

In this talk, we perform the first step towards proving Aboulker, Charbit, and Naserasr’s $\overrightarrow{\chi}$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.

We call an induced cycle of length at least four a hole. The parity of a hole is the parity of its length. Forbidding holes of certain types in a graph has deep structural implications. In 2006, Chudnovksy, Seymour, Robertson, and Thomas famously proved that a graph is perfect if and only if it does not contain an odd hole or a complement of an odd hole. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc provided a structural description of the class of even-hole-free graphs. I will describe the structure of all graphs that contain only holes of length $\ell$ for every $\ell \geq 7$ (joint work with Jake Horsfield, Myriam Preissmann, Paul Seymour, Ni Luh Dewi Sintiari, Cléophée Robin, Nicolas Trotignon, and Kristina Vuškovíc.

Analysis of how holes interact with graph structure has yielded detection algorithms for holes of various lengths and parities. In 1991, Bienstock showed it is NP-Hard to test whether a graph G has an even (or odd) hole containing a specified vertex $v \in V(G)$. In 2002, Conforti, Cornuéjols, Kapoor, and Vuškovíc gave a polynomial-time algorithm to recognize even-hole-free graphs using their structure theorem. In 2003, Chudnovsky, Kawarabayashi, and Seymour provided a simpler and slightly faster algorithm to test whether a graph contains an even hole. In 2019, Chudnovsky, Scott, Seymour, and Spirkl provided a polynomial-time algorithm to test whether a graph contains an odd hole. Later that year, Chudnovsky, Scott, and Seymour strengthened this result by providing a polynomial-time algorithm to test whether a graph contains an odd hole of length at least $\ell$ for any fixed integer $\ell \geq 5$. I will present a polynomial-time algorithm (joint work with Paul Seymour) to test whether a graph contains an even hole of length at least $\ell$ for any fixed integer $\ell \geq 4$.