Let $G=(V,E)$ be an undirected graph on n vertices and $\lambda :E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of integer labels (discrete times when the edge is present). Such a labelled graph $\mathcal{G}=(G,\lambda )$ is temporally connected if a path exists with non-decreasing times from every vertex to every other vertex. In a seminal paper, Kempe, Kleinberg, and Kumar [17] asked whether, given such a temporally connected graph, a sparse subset of edges always exists whose labels suffice to preserve temporal connectivity – a temporal spanner. Axiotis and Fotakis [5] answered negatively by exhibiting a family of $\mathrm{\Theta }(n^2)$-dense temporal graphs which admit no temporal spanner of density $o(n^2)$. In this paper, we give the first positive answer as to the existence of $o(n^2)$-sparse spanners in a dense class of temporal graphs, by showing (constructively) that if G is a complete graph, then one can always find a temporal spanner with $O(n\log n)$ edges.

让 $G=（\mathrm{伏特}，E）$是n个顶点上的无向图，并且$\lambda ：E\to {\mathrm{2个}}^{\mathbb{ñ}}$一个映射，为每个边缘分配一组非空的整数标签（存在边缘时的离散时间）。这样的标记图$\mathcal{G}=（G，\lambda ）$如果从每个顶点到每个其他顶点的路径都具有不减少的时间，则该路径在时间上是连通的。在一份开创性的论文中，Kempe，Kleinberg和Kumar [17]询问在给定这样的时间连接图的情况下，是否始终存在稀疏的边子集，它们的标签足以保留时间连接性，即时间扳手。Axiotis和Fotakis [5]展示了一个$\mathrm{\Theta }（{ñ}^{\mathrm{2个}}）$密度时间图，不包含密度的时间跨度 $Ø（{ñ}^{\mathrm{2个}}）$。在本文中，我们对存在的第一个肯定的答案$Ø（{ñ}^{\mathrm{2个}}）$在密集的一类时间图上稀疏生成扳手，方法是（建设性地）表明，如果G是一个完整的图，那么总可以找到一个具有以下内容的时间生成器：$Ø（ñ\mathrm{日志}ñ）$ 边缘。