A Logarithmic Bound for Simultaneous Embeddings of Planar Graphs

Abstract

A set $\mathcal {G}$ of planar graphs on the same number n of vertices is called simultaneously embeddable if there exists a set P of n points in the plane such that every graph $G \in \mathcal {G}$ admits a (crossing-free) straight-line embedding with vertices placed at points of P. A conflict collection is a set of planar graphs of the same order with no simultaneous embedding. A well-known open problem from 2007 posed by Brass, Cenek, Duncan, Efrat, Erten, Ismailescu, Kobourov, Lubiw and Mitchell, asks whether there exists a conflict collection of size 2. While this remains widely open, we give a short proof that for sufficiently large n there exists a conflict collection consisting of at most $(3+o(1))\log _2(n)$ planar graphs on n vertices. This significantly improves the previous exponential bound of $O(n\cdot 4^{n/11})$ for the same problem which was recently established by Goenka, Semnani and Yip.

We also give a computer-free proof that there exists a conflict collection of size 30, improving on the previously smallest known conflict collection of size 49 which was found using heavy computer assistance.