The Borsuk problem asks for the smallest number of subsets with strictly smaller diameters into which any bounded set in the $d$-dimensional space can be decomposed. It is a classical problem in combinatorial geometry that has been subject of much attention over the years, and research on variants of the problem continues nowadays in a plethora of directions. In this work, we propose a formulation of the problem in the context of graphs. Depending on how the graph is partitioned, we consider two different settings dealing either with the usual notion of diameter in abstract graphs, or with the diameter in the context of continuous graphs, where all points along the edges, instead of only the vertices, must be taken into account when computing distances. We present complexity results, exact computations and upper bounds on the parameters associated to the problem.

翻译：Borsuk问题要求找出将$d$维空间中任意有界集分解为直径严格更小的子集所需的最小子集数量。这是组合几何学中的经典问题，多年来备受关注，且至今仍沿诸多方向对其变体展开研究。本文提出了该问题在图论框架下的表述形式。根据图的分割方式，我们考虑了两种不同设定：其一是抽象图中通常意义上的直径概念；其二是连续图环境下的直径概念——在此设定下计算距离时需考虑边上的所有点，而不仅仅是顶点。我们给出了问题的复杂度结果、精确计算值以及相关参数的上界。