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

2026-01-02

AI Summary

This work investigates the existence of induced rainbow paths in triangle-free graphs equipped with a proper vertex coloring using $k$ colors. Specifically, it addresses whether such graphs necessarily contain an induced rainbow path of length $k$. By integrating tools from combinatorial graph theory, the probabilistic method, and extremal graph theory, the paper achieves two main advances: first, it substantially improves the known lower bound on the length of guaranteed induced rainbow paths from $(\log\log\log k)^{1/3 - o(1)}$ to $(\log k)^{1/2 - o(1)}$; second, it establishes that every such graph contains an induced path whose vertices collectively span at least $k/2$ distinct colors. These results significantly deepen the understanding of rainbow substructures in triangle-free graphs.

Technology Category

• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community
• Reasoning under Uncertainty: Graphical Models

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Responsible Web: Measurement, analysis, and circumvention of Web censorship
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• triangle-free graphs
• chromatic number
• induced rainbow path
• vertex coloring
• graph theory

Innovation

• induced rainbow path
• triangle-free graph
• chromatic number
• graph coloring
• logarithmic bound