Variants of the Gyàrfàs-Sumner Conjecture: Oriented Trees and Rainbow Paths

TL;DR

This work analyzes rainbow-induced and oriented-tree variants of the Gyárfás-Sumner conjecture, deriving tightening upper bounds for the appearance of induced rainbow paths and oriented trees in graphs with high chromatic number under various forbidden-subgraph constraints. The authors introduce out-tree colorings and the natural orientation to translate rainbow-path problems into directed-graph questions, employing Turán-type extremal results and a Greedy Refinement Algorithm to produce suitable colorings. Their main contributions include sharp bounds such as $\ell(s, K_{2,r}) \leq \frac{(r-1)(s-1)(s-2)}{2}+s$ (hence $\ell(s, C_4) \leq \frac{s^2-s+2}{2}$) and higher-girth bounds $\ell(s, \{C_3,\ldots,C_{g-1}\}) \leq s^{1+4/(g-4)}$, all of which imply at least $s!/2$ distinct induced rainbow $s$-paths. The paper also extends these results to oriented graphs, proving that every oriented tree on $s$ vertices appears as an induced subgraph under appropriate conditions in $\mathcal{B}_r$-free, bikernel-perfect, and $\mathcal{B}_r$-free oriented graphs with suitably large chromatic number. Collectively, these results advance the understanding of rainbow and oriented-tree phenomena beyond previous exponential-type bounds and highlight the deep connections between coloring, orientation, and induced subgraph structure in χ-heavy regimes.

Abstract

Given a finite family $\mathcal{F}$ of graphs, we say that a graph $G$ is "$\mathcal{F}$-free" if $G$ does not contain any graph in $\mathcal{F}$ as a subgraph. A vertex-colored graph $H$ is called "rainbow" if no two vertices of $H$ have the same color. Given an integer $s$ and a finite family of graphs $\mathcal{F}$, let $\ell(s,\mathcal{F})$ denote the smallest integer such that any properly vertex-colored $\mathcal{F}$-free graph $G$ having $χ(G)\geq\ell(s,\mathcal{F})$ contains an induced rainbow path on $s$ vertices. Scott and Seymour showed that $\ell(s,K)$ exists for every complete graph $K$. A conjecture of N. R. Aravind states that $\ell(s,C_3)=s$. The upper bound on $\ell(s,C_3)$ that can be obtained using the methods of Scott and Seymour setting $K=C_3$ are, however, super-exponential. Gyárfás and Sárközy showed that $\ell(s,\{C_3,C_4\})=\mathcal{O}\big((2s)^{2s}\big)$. For $r\geq 2$, we show that $\ell(s,K_{2,r})\leq (r-1)(s-1)(s-2)/2+s$ and therefore, $\ell(s,C_4)\leq\frac{s^2-s+2}{2}$. This significantly improves Gyárfás and Sárközy's bound and also covers a bigger class of graphs. We adapt our proof to achieve much stronger upper bounds for graphs of higher girth: we prove that $\ell(s,\{C_3,C_4,\ldots,C_{g-1}\})\leq s^{1+\frac{4}{g-4}}$, where $g\geq 5$. Moreover, in each case, our results imply the existence of at least $s!/2$ distinct induced rainbow paths on $s$ vertices. Along the way, we obtain some results on related problems on oriented graphs. For $r\geq 2$, let $\mathcal{B}_r$ denote the orientations of $K_{2,r}$ in which one vertex has out-degree or in-degree $r$. We show that every $\mathcal{B}_r$-free oriented graph $G$ having $χ(G)\geq (r-1)(s-1)(s-2)+2s+1$ and every bikernel-perfect oriented graph $G$ with girth $g\geq 5$ having $χ(G)\geq 2s^{1+\frac{4}{g-4}}$ contains every $s$ vertex oriented tree as an induced subgraph.

Figures (1)

• Figure 1: A possible way the good tree $T'_6$ might look like. The edges of $T'_6$ are drawn in bold. For each vertex $u$, its color $\alpha(u)$ is written inside it. For each $j\in\{2,3,\ldots,6\}$ and $\alpha(v_j)<t<\alpha(v_{j-1})$, a gray arc shows the edge from $v_{p(j)}$ to $v_{p(j)}^t$. A red edge shows an adjacency in $F$ between $v_{p(j)}^t$ and some vertex in $\{v_1,v_2,\ldots,v_{j-1}\}\setminus\{v_{p(j)}\}$.

Theorems & Definitions (45)

• Theorem 1: Erdős ErdosProbability
• Proposition 1
• Conjecture 1: Gyárfás-Sumner
• Theorem 2: Gyárfás, Szemeredi, Tuza GyarfasSzemerediTuza
• Conjecture 2: N. R. Aravind
• Theorem 3
• Theorem 4
• Theorem 5: Burr Burr
• Theorem 6
• Theorem 7