The crossing number of a graph G, denoted by cr(G), is defined as the smallest possible number of edge-crossings in a drawing of G in the plane. The cone CG, obtained from G by adding a new vertex adjacent to all the vertices in G. Let \(\phi _\mathrm {s}(k)=\min \{cr(CG)-cr(G)\}\), where the minimum is taken over all the simple graphs G with crossing number k. Alfaro et al. (SIAM J Discrete Math, 32: 2080–2093, 2018) determined \(\phi _\mathrm {s}(k)\le k\) for \(k\ge 3\), and proved \(\phi _\mathrm {s}(3)=3\), \(\phi _\mathrm {s}(4)=4\) and \(\phi _\mathrm {s}(5)=5\). In this work, we improve the upper bound for \(\phi _\mathrm {s}(k)\) and our main result includes the following, slightly surprising, fact: \(\phi _\mathrm {s}(6)=5\) and \(\phi _\mathrm {s}(7)=6\).

图G的交叉数，用cr ( G )表示，定义为平面中G图形中可能的最小边交叉数。锥体CG，得自ģ通过添加邻近于所有在顶点的新的顶点ģ。让\(\phi _\mathrm {s}(k)=\min \{cr(CG)-cr(G)\}\)，其中最小值取自所有具有交叉数k的简单图G。阿尔法罗等人。(SIAM J Discrete Math, 32: 2080–2093, 2018)为\(k\ge 3\)确定了\(\phi _\mathrm {s}(k)\le k \)，并证明了\(\phi _ \mathrm {s}(3)=3\), \(\phi _\mathrm {s}(4)=4\)和\(\phi _\mathrm {s}(5)=5\)。在这项工作中，我们改进了\(\phi _\mathrm {s}(k)\)的上限，我们的主要结果包括以下稍微令人惊讶的事实：\(\phi _\mathrm {s}(6 )=5\)和\(\phi _\mathrm {s}(7)=6\)。