Sparse graphs without long induced paths

Abstract

Graphs of bounded degeneracy are known to contain induced paths of order Ω(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 Ω((log⁡n)c) 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((log⁡log⁡n)2). We also show that the graphs we construct have linearly bounded coloring numbers.