We study the problem of finding a Hamiltonian cycle under the promise that the input graph has a minimum degree of at least $n/2$, where $n$ denotes the number of vertices in the graph. The classical theorem of Dirac states that such graphs (a.k.a. Dirac graphs) are Hamiltonian, i.e., contain a Hamiltonian cycle. Moreover, finding a Hamiltonian cycle in Dirac graphs can be done in polynomial time in the classical centralized model. This paper presents a randomized distributed CONGEST algorithm that finds w.h.p. a Hamiltonian cycle (as well as maximum matching) within $O(\log n)$ rounds under the promise that the input graph is a Dirac graph. This upper bound is in contrast to general graphs in which both the decision and search variants of Hamiltonicity require $\tilde{\Omega}(n^2)$ rounds, as shown by Bachrach et al. [PODC'19]. In addition, we consider two generalizations of Dirac graphs: Ore graphs and Rahman-Kaykobad graphs [IPL'05]. In Ore graphs, the sum of the degrees of every pair of non-adjacent vertices is at least $n$, and in Rahman-Kaykobad graphs, the sum of the degrees of every pair of non-adjacent vertices plus their distance is at least $n+1$. We show how our algorithm for Dirac graphs can be adapted to work for these more general families of graphs.

翻译：我们研究在输入图的最小度至少为$n/2$（$n$表示图的顶点数）的条件下寻找哈密顿圈的问题。经典狄拉克定理表明此类图（即Dirac图）是哈密顿图，即包含哈密顿圈。此外，在经典集中式计算模型中，可在多项式时间内找到Dirac图中的哈密顿圈。本文提出一种随机化分布式CONGEST算法，在输入图为Dirac图时，能以高概率在$O(\log n)$轮内找到哈密顿圈（以及最大匹配）。这一上界与一般图形成鲜明对比：如Bachrach等人[PODC'19]所示，哈密顿性的判定与搜索变体在一般图中均需要$\tilde{\Omega}(n^2)$轮。此外，我们考虑Dirac图的两种推广：Ore图和Rahman-Kaykobad图[IPL'05]。在Ore图中，每对非相邻顶点的度数之和至少为$n$；在Rahman-Kaykobad图中，每对非相邻顶点的度数之和加上其距离至少为$n+1$。我们展示了如何将Dirac图的算法适配至这些更一般的图族。