Long induced paths in graphs

TL;DR

This paper proves new lower bounds on the size of induced paths in planar and related graphs, showing that such paths grow logarithmically with the number of vertices, and explores conjectures for broader graph classes.

Contribution

It establishes the best possible logarithmic lower bounds for induced paths in 3-connected planar graphs and extends results to graphs on fixed surfaces and interval graphs.

Findings

Induced paths of size Ω(log n) exist in 3-connected planar graphs.

Planar graphs with a long path also contain induced paths of size Ω(√log n).

The paper proves the conjecture for interval graphs.

Abstract

We prove that every 3-connected planar graph on $n$ vertices contains an induced path on $\Omega(\log n)$ vertices, which is best possible and improves the best known lower bound by a multiplicative factor of $\log \log n$. We deduce that any planar graph (or more generally, any graph embeddable on a fixed surface) with a path on $n$ vertices, also contains an induced path on $\Omega(\sqrt{\log n})$ vertices. We conjecture that for any $k$, there is a contant $c(k)$ such that any $k$-degenerate graph with a path on $n$ vertices also contains an induced path on $\Omega((\log n)^{c(k)})$ vertices. We provide examples showing that this order of magnitude would be best possible (already for chordal graphs), and prove the conjecture in the case of interval graphs.