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arXiv · 2409.01678

Subgraph-universal planar graphs for trees

Helena Bergold·Vesna Iršič·Robert Lauff·Joachim Orthaber·Manfred Scheucher·Alexandra Wesolek
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Abstract

We show that there exists an outerplanar graph on $O(n^{c})$ vertices for $c = \log_2(3+\sqrt{10}) \approx 2.623$ that contains every tree on $n$ vertices as a subgraph. This extends a result of Chung and Graham from 1983 who showed that there exist (non-planar) $n$-vertex graphs with $O(n \log n)$ edges that contain all trees on $n$ vertices as subgraphs and a result from Gol'dberg and Livshits from 1968 who showed that there exists a universal tree for $n$-vertex trees on $n^{O(\log(n))}$ vertices. Furthermore, we determine the number of vertices needed in the worst case for a planar graph to contain three given trees as subgraph to be on the order of $\frac{3}{2}n$, even if the three trees are caterpillars. This answers a question recently posed by Alecu et al. in 2024. Lastly, we investigate (outer)planar graphs containing all (outer)planar graphs as subgraph, determining exponential lower bounds in both cases. We also construct a planar graph on $n^{O(\log(n))}$ vertices containing all $n$-vertex outerplanar graphs as subgraphs.

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Helena Bergold, Vesna Iršič, Robert Lauff, Joachim Orthaber, Manfred Scheucher, Alexandra Wesolek. 2024-09-03. Subgraph-universal planar graphs for trees. https://arxiv.org/abs/2409.01678

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