The graph of diameters on a given subset \(V\) of a metric space is defined as the graph with vertices the elements of \(V\), where two vertices are adjacent if they are at distance equal to the diameter of \(V\). It is proved that any two cycles of odd order in the graph of diameters in the 3-dimensional Euclidean space have a common vertex. Some properties of such graphs in Banach spaces are also studied.