Log In Sign Up

Plane Spanning Trees in Edge-Colored Simple Drawings of K_n

by   Oswin Aichholzer, et al.

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 ⌈ (n+5)/6⌉-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.)


page 1

page 2

page 3

page 4


Shooting Stars in Simple Drawings of K_m,n

Simple drawings are drawings of graphs in which two edges have at most o...

Simple Compact Monotone Tree Drawings

A monotone drawing of a graph G is a straight-line drawing of G such tha...

The Partition Spanning Forest Problem

Given a set of colored points in the plane, we ask if there exists a cro...

Drawing Graphs as Spanners

We study the problem of embedding graphs in the plane as good geometric ...

Ordered Level Planarity, Geodesic Planarity and Bi-Monotonicity

We introduce and study the problem Ordered Level Planarity which asks fo...

Searching by Heterogeneous Agents

In this work we introduce and study a pursuit-evasion game in which the ...

Angle-monotone Paths in Non-obtuse Triangulations

We reprove a result of Dehkordi, Frati, and Gudmundsson: every two verti...