A quasi-optimal upper bound for induced paths in sparse graphs

TL;DR

This paper establishes an almost tight upper bound on the length of induced paths in 2-degenerate graphs with long paths, refining previous bounds for sparse graphs.

Contribution

It provides a nearly matching upper bound for the maximum induced path length in 2-degenerate graphs with long paths, advancing understanding of sparse graph structures.

Findings

Constructed 2-degenerate graphs with long paths and small induced paths

Established an upper bound of O((log log n)^{1+o(1)}) for induced path length

Refined previous bounds for graphs of bounded degeneracy

Abstract

In 2012, Ne\v{s}et\v{r}il and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $\Omega(\log \log n)$. In this paper we give an almost matching upper bound by describing, for arbitrarily large values of $n$, 2-degenerate graphs that have a path of order $n$ and where the longest induced paths have order $O((\log \log n)^{1+o(1)})$.