On RAC drawings of graphs with one bend per edge

A k-bend right-angle-crossing drawing (or k-bend RAC drawing, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are \(90^\circ \). Accordingly, a graph that admits a \(k\)-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for short). In this paper, we continue the study of the maximum edge-density of \(1\)-bend RAC graphs. We show that an n-vertex \(1\)-bend RAC graph cannot have more than \(5.5n-O(1)\) edges. We also demonstrate that there exist infinitely many n-vertex \(1\)-bend RAC graphs with exactly \(5n-O(1)\) edges. Our results improve both the previously known best upper bound of \(6.5n-O(1)\) edges and the corresponding lower bound of \(4.5n-O(\sqrt{n})\) edges by Arikushi et al. (Comput. Geom. 45(4), 169–177 (2012)).

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