Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Tuesday, December 15, 2020, 12:15 pm

Duration: 30 minutes

Location: Zoom: conference room

Speaker: Domagoj Bradac

Powers of Hamilton cycles of high discrepancy are unavoidable

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 many more edges in one of the colors. 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 matches the minimum degree requirement of the Pósa-Seymour conjecture.

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