A Logarithmic Bound for Simultaneous Embeddings of Planar Graphs

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 constitutes a double-exponential improvement over the previously best known bound of \(O(n\cdot 4^{n/11})\) for the same problem by Goenka et al. (Graphs Combin 39:100, 2023). Using our method we also provide a computer-free proof that for every integer \(n\in [107,193]\) there exists a conflict collection of 30 planar n-vertex graphs, improving upon the previously smallest known conflict collection consisting of 49 graphs of order 11, which was found using heavy computer assistance. While the construction by Goenka et al. was explicit, our construction of a conflict collection of size \(O(\log n)\) is based on the probabilistic method and is thus only implicit. Motivated by this, for every large enough n we give a different, fully explicit construction of a collection of less than \(n^6\) planar n-vertex graphs with no simultaneous embedding.