Towards a conjecture on long induced rainbow paths in triangle-free graphs

TL;DR

This paper advances the understanding of rainbow paths in triangle-free graphs by establishing new lower bounds on the length of induced rainbow paths relative to the chromatic number, and demonstrates the existence of paths covering half the colors.

Contribution

It improves previous bounds on the length of induced rainbow paths in triangle-free graphs and proves the existence of paths that encompass half the available colors.

Findings

Induced rainbow path length improved to $( ext{log }k)^{1/2 - o(1)}$ vertices.

Existence of an induced path seeing half of the colors in the graph.

Progress towards the conjecture on long induced rainbow paths in triangle-free graphs.

Abstract

Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $\phi$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $\phi$. Scott and Seymour proved the existence of an induced rainbow path on $(\log \log \log k)^{\frac{1}{3}- o(1)}$ vertices. We improve this to $(\log k)^{\frac{1}{2}- o(1)}$ vertices. Further, we prove the existence of an induced path that sees $\frac{k}{2}$ colors.