Powers of Hamilton cycles of high discrepancy are unavoidable

TL;DR

This paper establishes the minimum degree conditions necessary to find high-discrepancy powers of Hamilton cycles in edge-colored graphs, extending classical Hamiltonicity results to a discrepancy setting.

Contribution

It determines the degree threshold for large discrepancy Hamilton cycle powers, answering a question about the interplay between Hamiltonicity and discrepancy in colored graphs.

Findings

Minimum degree threshold for large discrepancy Hamilton cycle powers identified

Threshold for r≥3 matches Posa-Seymour conjecture requirements

Answers a previously open question in graph discrepancy theory

Abstract

The P\'osa-Seymour conjecture asserts that every graph on $n$ vertices with minimum degree at least $(1 - 1/(r+1))n$ contains the $r^{th}$ power of a Hamilton cycle. Koml\'os, S\'ark\"ozy and Szemer\'edi famously proved the conjecture for large $n.$ The notion of discrepancy appears in many areas of mathematics, including graph theory. In this setting, a graph $G$ is given along with a $2$-coloring of its edges. One is then asked to find in $G$ a copy of a given subgraph with a large discrepancy, i.e., with significantly more than half of its edges in one color. For $r \geq 2,$ we determine the minimum degree threshold needed to find the $r^{th}$ power of a Hamilton cycle of large discrepancy, answering a question posed by Balogh, Csaba, Pluh\'ar and Treglown. Notably, for $r \geq 3,$ this threshold approximately matches the minimum degree requirement of the P\'osa-Seymour conjecture.