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 $24n-26$ 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,对于任意$n$顶点图,若其在平面内可被绘制为每边两折的RAC图(即每边为具有两折的多段线,且任意两边仅能以直角相交),则当$n\geq 3$时,其边数至多为$24n-26$。此结果改进了之前$74.2n$的上界,是12年多来首次突破。证明的关键要素在于对具有多段线边(其首尾线段平行或正交)的平面多重图规模上界的确立。