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arXiv · 2304.02164

Improved upper bounds on even-cycle creating Hamilton paths

John Byrne·Michael Tait
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Abstract

We study the function $H_n(C_{2k})$, the maximum number of Hamilton paths such that the union of any pair of them contains $C_{2k}$ as a subgraph. We give upper bounds on this quantity for $k\ge 3$, improving results of Harcos and Solt\'esz, and we show that if a conjecture of Ustimenko is true then one additionally obtains improved upper bounds for all $k\geq 6$. {We also give bounds on $H_n(K_{2,3})$ and $H_n(K_{2,4})$. In order to prove our results, we extend a theorem of Krivelevich which counts Hamilton cycles in $(n, d, \lambda)$-graphs to bipartite or irregular graphs, and then apply these results to generalized polygons and the constructions of Lubotzky-Phillips-Sarnak and F\"uredi.

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John Byrne, Michael Tait. 2023-04-04. Improved upper bounds on even-cycle creating Hamilton paths. https://arxiv.org/abs/2304.02164

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