Sparse graphs without long induced paths

Abstract

Graphs of bounded degeneracy are known to contain induced paths of order $\Omega (\log \log n)$ when they contain a path of order n, as proved by Nešetřil and Ossona de Mendez (2012). In 2016 Esperet, Lemoine, and Maffray conjectured that this bound could be improved to $\Omega \left({(\log n)}^c\right)$ for some constant $c>0$ depending on the degeneracy.

We disprove this conjecture by constructing, for arbitrarily large values of n, a graph that is 2-degenerate, has a path of order n, and where all induced paths have order $O\left({(\log \log n)}^2\right)$. We also show that the graphs we construct have linearly bounded coloring numbers.