Tight bound for powers of Hamilton cycles in tournaments

Abstract

A basic result in graph theory says that any n-vertex tournament with in-and out-degrees larger than n−2 4 contains a Hamilton cycle, and this is tight. In 1990, Bollobás and Häggkvist significantly extended this by showing that for any fixed k and ε > 0, and sufficiently large n, all tournaments with degrees at least n 4 + εn contain the k-th power of a Hamilton cycle. Given this, it is natural to ask for a more accurate error term in the degree condition, and also how large n should be in the Bollobás-Häggkvist theorem. We essentially resolve both questions. First, we show that if the degrees are at least n 4 + cn 1−1/⌈k/2⌉ for some constant c = c(k), then the tournament contains the k-th power of a Hamilton cycle. In particular, in order to guarantee the square of a Hamilton cycle, one only needs a constant additive term. We also present a construction which, modulo a well-known conjecture on Turán numbers for complete bipartite graphs, shows that the error term must be of order at least n 1−1/⌈(k−1)/2⌉ , which matches our upper bound for all even k. For odd k, we believe that the lower bound can be improved. Indeed, we show that for k = 3, there are tournaments with degrees n 4 +Ω(n 1/5 ) and no cube of a Hamilton cycle. In addition, our results imply that the Bollobás-Häggkvist theorem already holds for n = ε −Θ(k) , which is best possible.

Powers of paths and cycles in tournaments

“…An old result of Bollobás and Häggkvist [2] says that for every k and ε > 0, any tournament on sufficiently many vertices with minimum semi-degree at least (1/4 + ε)n contains a k-th power of a Hamilton cycle and this is tight up to the o(1) error term. Recently, Draganic, Munhá Correia, and Sudakov [6] were able to find an almost tight bound for the error term.…”