Non-crossing monotone paths and binary trees in edge-ordered complete geometric graphs

Abstract

An edge-ordered graph is a graph with a total ordering of its edges. A path $P=v_1v_2\ldots v_k$ in an edge-ordered graph is called increasing if $(v_iv_{i+1}) < (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$; and it is called decreasing if $(v_iv_{i+1}) > (v_{i+1}v_{i+2})$ for all $i = 1,\ldots,k-2$. We say that P is monotone if it is increasing or decreasing. A rooted tree T in an edge-ordered graph is called monotone if either every path from the root to a leaf is increasing or every path from the root to a leaf is decreasing.

Let G be a graph. In a straight-line drawing D of G, its vertices are drawn as different points in the plane and its edges are straight line segments. Let $\overline{\alpha}(G)$ be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing path of length $\overline{\alpha}(G)$. Let $\overline{\tau}(G)$ be the largest integer such that every edge-ordered straight-line drawing of G contains a monotone non-crossing complete binary tree of $\overline{\tau}(G)$ edges. In this paper we show that $\overline \alpha(K_n) = \Omega(\log\log n)$, $\overline \alpha(K_n) = O(\log n), \overline \tau(K_n) = \Omega(\log\log \log n)$ and $\overline \tau(K_n) = O(\sqrt{n \log n})$.