Abstract
Károlyi, Pach, and Tóth proved that every 2-edge-colored straight-line drawing of the complete graph contains a monochromatic plane spanning tree. It is open if this statement generalizes to other classes of drawings, specifically, to simple drawings of the complete graph. These are drawings where edges are represented by Jordan arcs, any two of which intersect at most once. We present two partial results towards such a generalization. First, we show that the statement holds for cylindrical simple drawings. (In a cylindrical drawing, all vertices are placed on two concentric circles and no edge crosses either circle.) Second, we introduce a relaxation of the problem in which the graph is k-edge-colored, and the target structure must be hypochromatic, that is, avoid (at least) one color class. In this setting, we show that every \(\lceil (n+5)/6\rceil \)-edge-colored monotone simple drawing of \(K_n\) contains a hypochromatic plane spanning tree. (In a monotone drawing, every edge is represented as an x-monotone curve.)
We are particularly grateful to Irene Parada for bringing this problem to our attention. We also thank the organizers of the \(4^{th}\) DACH Workshop on Arrangements, that took place in February 2020 in Malchow and was funded by Deutsche Forschungsgemeinschaft (DFG), the Austrian Science Fund (FWF) and the Swiss National Science Foundation (SNSF). M. H. is supported by SNSF Project 200021E-171681. R. P. and A. W. are supported by FWF grant W1230. J. O. is supported by ERC StG 757609. N. S. is supported by DFG Project MU3501/3-1. D. P. and B. V. are supported by FWF Project I 3340-N35.
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Notes
- 1.
Note that the coloring need not be proper nor have any other special properties.
- 2.
Two simple drawings of \(K_n\) are weakly isomorphic iff they have the same crossing edge pairs.
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Aichholzer, O. et al. (2020). Plane Spanning Trees in Edge-Colored Simple Drawings of \(K_{n}\). In: Auber, D., Valtr, P. (eds) Graph Drawing and Network Visualization. GD 2020. Lecture Notes in Computer Science(), vol 12590. Springer, Cham. https://doi.org/10.1007/978-3-030-68766-3_37
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