Improved upper bounds on even-cycle creating Hamilton paths

We study the function H n ( C 2 k ) , the maximum number of Hamilton paths such that the union of any pair of them contains C 2 k as a subgraph. We give upper bounds on this quantity for k ≥ 3 , improving results of Harcos and Soltész, and we show that if a conjecture of Ustimenko is true then one additionally obtains improved upper bounds for all k ≥ 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 , λ ) -graphs to bipartite or irregular graphs, and then apply these results to generalized polygons and the constructions of Lubotzky-Phillips-Sarnak and Füredi.