Log In Sign Up

Long induced paths in minor-closed graph classes and beyond

by   Claire Hilaire, et al.

In this paper we show that every graph of pathwidth less than k that has a path of order n also has an induced path of order at least 1/3 n^1/k. This is an exponential improvement and a generalization of the polylogarithmic bounds obtained by Esperet, Lemoine and Maffray (2016) for interval graphs of bounded clique number. We complement this result with an upper-bound. This result is then used to prove the two following generalizations: - every graph of treewidth less than k that has a path of order n contains an induced path of order at least 1/4 (log n)^1/k; - for every non-trivial graph class that is closed under topological minors there is a constant d ∈ (0,1) such that every graph from this class that has a path of order n contains an induced path of order at least (log n)^d. We also describe consequences of these results beyond graph classes that are closed under topological minors.


page 1

page 2

page 3

page 4


Shifting paths to avoidable ones

An extension of an induced path P in a graph G is an induced path P' suc...

Treewidth versus clique number in graph classes with a forbidden structure

Treewidth is an important graph invariant, relevant for both structural ...

The Erdős-Hajnal conjecture for caterpillars and their complements

The celebrated Erdős-Hajnal conjecture states that for every proper here...

Avoidability beyond paths

The concept of avoidable paths in graphs was introduced by Beisegel, Chu...

Notes on Graph Product Structure Theory

It was recently proved that every planar graph is a subgraph of the stro...

Tangled Paths: A Random Graph Model from Mallows Permutations

We introduce the random graph 𝒫(n,q) which results from taking the union...

Separating the edges of a graph by a linear number of paths

Recently, Letzter proved that any graph of order n contains a collection...