A temporal graph can be represented by a graph with an edge labelling, such that an edge is present in the network if and only if the edge is assigned the corresponding time label. A journey is a labelled path in a temporal graph such that labels on successive edges of the path are increasing, and if all vertices admit journeys to all other vertices, the temporal graph is temporally connected. A temporal spanner is a sublabelling of the temporal graph such that temporal connectivity is maintained. The study of temporal spanners has raised interest since the early 2000's. Essentially two types of studies have been conducted: the positive side where families of temporal graphs are shown to (deterministically or stochastically) admit sparse temporal spanners, and the negative side where constructions of temporal graphs with no sparse spanners are of importance. Often such studies considered temporal graphs with happy or simple labellings, which associate exactly one label per edge. In this paper, we focus on the negative side and consider proper labellings, where multiple labels per edge are allowed. More precisely, we aim to construct dense temporally connected graphs such that all labels are necessary for temporal connectivity. Our contributions are multiple: we present the first labellings maximizing a local density measure; exact or asymptotically tight results for basic graph families, which are then extended to larger graph families; an extension of an efficient temporal graph labelling generator; and overall denser labellings than previous work even when restricted to happy labellings.

翻译：时序图可表示为带有边标记的图，其中边存在当且仅当该边被分配了对应的时间标签。旅程是时序图中标签递增的标记路径，若所有顶点都能通过旅程到达其他所有顶点，则该时序图称为时序连通的。时序生成子图是时序图的子标记，使得时序连通性得以保持。自21世纪初以来，时序生成子图的研究引起了广泛关注。相关研究主要分为两类：正面研究展示某些时序图族（确定性地或随机地）具有稀疏的时序生成子图；负面研究则关注构建不存在稀疏生成子图的时序图。这类研究通常考虑快乐标记或简单标记的时序图，其中每条边仅关联一个标签。本文聚焦负面研究，并考虑允许每条边具有多个标签的恰当标记。具体而言，我们旨在构建密集的时序连通图，使得所有标签对于时序连通性均为必要。我们的贡献包括：提出首个最大化局部密度度量的标记方案；给出基本图族的精确或渐近紧致结果，并推广至更大图族；扩展高效的时序图标记生成器；即使在限制为快乐标记的情况下，也获得比以往研究更密集的标记方案。