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 > 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 > 3, this threshold approximately matches the minimum degree requirement of the Pósa-Seymour conjecture.
Publisher
The Electronic Journal of Combinatorics
Subject
Computational Theory and Mathematics,Geometry and Topology,Theoretical Computer Science,Applied Mathematics,Discrete Mathematics and Combinatorics
Cited by
2 articles.
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1. Discrepancies of Subtrees;AIRO Springer Series;2024
2. Oriented discrepancy of Hamilton cycles;Journal of Graph Theory;2023-02-26