Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

N. R. Aravind, Shiwali Gupta, Rogers Mathew

TL;DR

The paper addresses the problem of long induced rainbow paths in triangle-free graphs with a proper coloring $\\phi$, aiming toward the conjecture that every colored triangle-free graph contains an induced rainbow path on $\\chi(G)$ vertices. Building on Scott and Seymour's bound, the authors refine the coloring-grading framework and select $r = 4 \\cdot 2^{2(s-1)\\log(s-1)}$ to derive a dichotomy that yields an induced rainbow path on $s$ vertices or enables its construction; directed-graph decompositions and BFS arguments then yield a path of length $(\\log \\chi(G))^{1/2 - o(1)}$. Additionally, they prove that for every vertex there exists an induced $\\tfrac{\\chi(G)}{2}$-colorful path starting at that vertex via an induction on $\\chi(G)$. These results advance the conjecture in the general, girth-unrestricted setting by providing stronger long induced rainbow paths and colorful variants.

Abstract

Given a triangle-free graph $G$ with chromatic number $k$ and a proper vertex coloring $φ$ of $G$, it is conjectured that $G$ contains an induced rainbow path on $k$ vertices under $φ$. 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.

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

TL;DR

The paper addresses the problem of long induced rainbow paths in triangle-free graphs with a proper coloring , aiming toward the conjecture that every colored triangle-free graph contains an induced rainbow path on vertices. Building on Scott and Seymour's bound, the authors refine the coloring-grading framework and select to derive a dichotomy that yields an induced rainbow path on vertices or enables its construction; directed-graph decompositions and BFS arguments then yield a path of length . Additionally, they prove that for every vertex there exists an induced -colorful path starting at that vertex via an induction on . These results advance the conjecture in the general, girth-unrestricted setting by providing stronger long induced rainbow paths and colorful variants.

Abstract

Given a triangle-free graph with chromatic number and a proper vertex coloring of , it is conjectured that contains an induced rainbow path on vertices under . Scott and Seymour proved the existence of an induced rainbow path on vertices. We improve this to vertices. Further, we prove the existence of an induced path that sees colors.
Paper Structure (2 sections, 4 theorems, 1 equation)

This paper contains 2 sections, 4 theorems, 1 equation.

Table of Contents

  1. Introduction
  2. Our results

Key Result

Lemma 1

Let $s \ge 3$ be an integer and let $r=4 \cdot 2^{2(s-1)\log (s-1)}$. Let $(G, \phi)$ be a colored graph with $\chi(G) \ge k\cdot r$, where $k$ is some positive integer. Let $(W_1, W_2, \dots, W_n)$ be a $k$-colorable grading of $G$. Then, one of the following statements always holds: (i) $G$ contai

Theorems & Definitions (8)

  • Conjecture 1
  • Lemma 1
  • proof
  • Lemma 2
  • Remark
  • Theorem 1
  • Theorem 2
  • proof