Powers of Hamilton cycles of high discrepancy are unavoidable

Abstract

The Pósa-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. Komlós, Sárközy and Szemerédi 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 $r\ge 2,$ 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, Pluhár and Treglown. Notably, for $r\ge 3,$ this threshold approximately matches the minimum degree requirement of the Pósa-Seymour conjecture.