Plane Spanning Trees in Edge-Colored Simple Drawings of $K_n$

TL;DR

This paper investigates the existence of monochromatic or hypochromatic plane spanning trees in various classes of edge-colored simple drawings of the complete graph, extending known results and introducing new partial results and relaxations.

Contribution

It proves the existence of monochromatic plane spanning trees in cylindrical simple drawings and introduces a new hypochromatic variant with bounds in monotone simple drawings.

Findings

Monochromatic plane spanning trees exist in cylindrical simple drawings.

Every rac{n+5}{6}rac{n+5}{6}-edge-colored monotone simple drawings contain a hypochromatic plane spanning tree.

Partial progress towards generalizing known results to broader classes of simple graph drawings.

Abstract

K\'{a}rolyi, Pach, and T\'{o}th 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.)