Proving a directed analogue of the Gyárfás-Sumner conjecture for orientations of $P_4$

TL;DR

The paper advances the directed Gyárfás–Sumner program by proving that every orientation of $P_4$ is $\overrightarrow{\chi}$-bounding, i.e., $H$-free oriented graphs have a bounded dichromatic number in terms of their clique number when $H$ is an orientation of $P_4$. The authors develop a dipolar-set framework anchored by closed-tournament structures and path-minimizing extensions, enabling tight control of the dichromatic number via induction on the clique number $\omega$. They give explicit, orientation-dependent bounds, e.g., $\overrightarrow{\chi}(D)\le(\omega+3)^{\omega+4.5}$ in the $\overrightarrow{Q_4}$-free case, $\le(\omega+6)^{\omega+7.5}$ for the $\overrightarrow{P_4}$-free case, and $\le(\omega+7)^{\omega+8.5}$ for the $\overrightarrow{A_4}$-free case, culminating in a unified bound $\overrightarrow{\chi}(D)\le(\omega(D)+7)^{(\omega(D)+8.5)}$ for all orientations of $P_4$. This work combines structural decompositions of dipolar sets, strong neighborhood arguments, and careful partitioning of forward-induced paths to bound the dichromatic number in terms of the underlying clique size. It represents the first solid step toward resolving ACN's $\overrightarrow{\chi}$-boundedness for orientations of trees beyond stars and strengthens the toolkit for heroic-set analyses in directed graphs. The results open avenues for exploring extensions to longer oriented paths, with open questions about potential polynomial bounds and the role of transitive tournaments in broader classes.

Abstract

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 $ω(G)$ has chromatic number at most $f(ω(G))$. 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 $\overrightarrowχ$-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 $f(ω(D))$, where $ω(D)$ is the size of a maximum clique in the graph underlying $D$. In this paper, we perform the first step towards proving Aboulker, Charbit, and Naserasr's $\overrightarrowχ$-boundedness conjecture by showing that it holds when $F$ is any orientation of a path on four vertices.

Figures (5)

• Figure 1: An illustration of the extension of a closed tournament $C$ into the \ref{['def:niceset']} set $N[C \cup X]$. Highlighted in blue, $Z$ consists of neighbors of $C$ that are not strong, i.e., do not have both an in-neighbor and an out-neighbor in $C$. The set $X$ consists of the strong neighborhood of $C$, while set $Y$ contains all neighbors of $X$ not in $N[C]$. Note that arcs between $Z$ and $X$ or $Y$ are not represented here. In Lemma \ref{['lem:buildinganiceset']}, we prove that if there is some vertex in $N[C \cup X]$ with both an in-neighbor and an out-neighbor in the rest of the oriented graph (drawn in dashed red), then $N[C \cup X]$ contains all orientations of $P_4$ as an induced oriented subgraph.
• Figure 2: Vertex sets $Q,R,S$, such that no arc lies between $Q$ and $S$, the vertices in $R$ are all \ref{['def:strongnbd']} of $Q$ and $S$ is a subset of neighbors of $R$. We illustrate the case of graphs forbidding \ref{['def:p4']}. Other orientations behave symmetrically. In dark green, a vertex $r \in R$ is depicted with an out-neighbor $s \in S$. Then if $s$ has an out-neighbor $s_1 \in S \setminus N(r)$ there would be an induced \ref{['def:p4']}, a contradiction. Symmetrically in dark blue, an in-neighbor $s' \in S$ of $r$ cannot admit an in-neighbor $s_2' \in S \backslash N(r)$.
• Figure 3: A shortest path directed path $P=p_1 \to ... \to p_{\ell}$ along with the partition $(F_1,...,F_{\ell})$ of $N(P)$ by first attachment on $P$. Note that the setting is symmetric for the partition of $N(P)$ by the last attachment. Each class of the partition $F_i$ is represented as a circle and further split into $F_i^+$ in green and $F_i^-$ in blue, all possible arcs towards $P$ are drawn in gray. An arc from $F_i^-$ to $F_j$ with $j > i$ would induce a \ref{['def:p4']} using $(p_{i-1},p_i)$, as highlighted in dark blue. An arc from $F_i^+$ to $F_j$ would induce a \ref{['def:a4']}, represented in dark green.
• Figure 4: On the bottom, a shortest path $P$ in $D$, and an arc $(w,v)$ between neighbors of $P$ in $R^p$. Illustrated here is the case where the last neighbor $p_{w_2}$ of $w$ on $P$ appears just before the first neighbor $p_{v_1}$ of $v$, all possible arcs are shown in gray. Then, path $p_{w_2-1},p_{w_2},p_{v_1},p_{v_1 + 1}$ cannot induce a \ref{['def:p4']}. Any arc possibly preventing this, shown in dash-dotted green, yields a path from $v$ to $w$ of length at most five.
• Figure 5: A shortest directed path $P$, along with $v \in F_i \setminus W^a$ and $w \in F_j \setminus W^a$ of \ref{['def:ra']}. Since vertex $w \notin W^a$, it must also belong to some $L_x^+$ with $x > j$, meaning its last neighbor on $P$ is an out-neighbor. Then, since $x>j>i+2$, an arc $(v,w)$ would yield a shorter path from $p_1$ to $p_{\ell}$.

Theorems & Definitions (40)

• Conjecture 1.1: The Gyárfás-Sumner conjecture gyarfasConjecturesumnerConjecture
• Conjecture 1.2: The ACN $\overrightarrow{\chi}$-boundedness conjecture AlboukerGyarfasSumner4Digraphs2020
• Theorem 1.2
• Definition 1.3: path-minimizing closed tournament
• Definition 2.3: dipolar set
• Lemma 2.4: Lemma 17 in AboulkerChordalDirectedGraphsAreNotChiBounded2022
• Lemma 3.1
• proof
• proof
• proof