It is shown that every $n$-vertex graph that admits a 2-bend RAC drawing in the plane, where the edges are polylines with two bends per edge and any pair of edges can only cross at a right angle, has at most $20n-24$ edges for $n\geq 3$. This improves upon the previous upper bound of $74.2n$; this is the first improvement in more than 12 years. A crucial ingredient of the proof is an upper bound on the size of plane multigraphs with polyline edges in which the first and last segments are either parallel or orthogonal.
翻译:研究表明,每条边具有两个折点且任意两条边仅能以直角交叉的平面2-折RAC绘制图中,对于顶点数n≥3的n顶点图,其边数至多为20n-24。该结果将此前74.2n的上界显著改进,这是12年多来首次实现突破。证明的关键要素是建立一类平面多重图的最大规模上界,这类图的折线边首尾线段要么平行要么正交。