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,任意在平面上允许2弯曲RAC绘图的$n$顶点图(其中边为每条边含两个折点折线,且任意两对边仅能以直角相交),当$n\geq 3$时最多有$20n-24$条边。这一结果改进了此前$74.2n$的上界,也是12年多来的首次突破。证明的关键要素是给出了平面多重图大小的一个上界,这类图以折线为边,且其首尾线段要么平行要么正交。