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arXiv · 2203.07148

Oriented discrepancy of Hamilton cycles

Lior Gishboliner·Michael Krivelevich·Peleg Michaeli
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Abstract

We propose the following conjecture extending Dirac's theorem: if $G$ is a graph with $n\ge 3$ vertices and minimum degree $\delta(G)\ge n/2$, then in every orientation of $G$ there is a Hamilton cycle with at least $\delta(G)$ edges oriented in the same direction. We prove an approximate version of this conjecture, showing that minimum degree $n/2 + O(k)$ guarantees a Hamilton cycle with at least $(n+k)/2$ edges oriented in the same direction. We also study the analogous problem for random graphs, showing that if the edge probability $p = p(n)$ is above the Hamiltonicity threshold, then, with high probability, in every orientation of $G \sim G(n,p)$ there is a Hamilton cycle with $(1-o(1))n$ edges oriented in the same direction.

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Lior Gishboliner, Michael Krivelevich, Peleg Michaeli. 2022-03-14. Oriented discrepancy of Hamilton cycles. https://arxiv.org/abs/2203.07148

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