Long induced paths in sparse graphs and graphs with forbidden patterns

TL;DR

This paper investigates conditions under which sparse graphs and ordered graphs contain long induced paths, providing new bounds and simplified proofs for the existence of such paths when certain subgraphs are forbidden.

Contribution

It improves lower bounds on induced path lengths in graphs forbidding complete bipartite subgraphs and introduces a unified approach for ordered graphs with forbidden subgraphs.

Findings

Induced paths of length $( ext{log} ext{log} n)^{1/5-o(1)}$ exist in graphs forbidding $K_{t,t}$.

Simplified proofs for long induced paths in ordered graphs with forbidden matchings.

Long induced paths of size at least $ ext{Omega}(( ext{log} ext{log} ext{log} n)^{1/3})$ can be found in certain ordered graphs.

Abstract

Consider a graph $G$ with a path $P$ of order $n$. What conditions force $G$ to also have a long induced path? As complete bipartite graphs have long paths but no long induced paths, a natural restriction is to forbid some fixed complete bipartite graph $K_{t,t}$ as a subgraph. In this case we show that $G$ has an induced path of order $(\log \log n)^{1/5-o(1)}$. This is an exponential improvement over a result of Galvin, Rival, and Sands (1982) and comes close to a recent upper bound of order $O((\log \log n)^2)$. Another way to approach this problem is by viewing $G$ as an ordered graph (where the vertices are ordered according to their position on the path $P$). From this point of view it is most natural to consider which ordered subgraphs need to be forbidden in order to force the existence of a long induced path. Focusing on the exclusion of ordered matchings, we improve or recover a number of existing results with much simpler proofs, in a unified way. We also show that if some forbidden ordered subgraph forces the existence of a long induced path in $G$, then this induced path has size at least $\Omega((\log \log \log n)^{1/3})$, and can be chosen to be increasing with respect to $P$.