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Long induced paths in sparse graphs and graphs with forbidden patterns

Julien Duron, Louis Esperet, Jean-Florent Raymond

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 $Ω((\log \log \log n)^{1/3})$, and can be chosen to be increasing with respect to $P$.

Long induced paths in sparse graphs and graphs with forbidden patterns

Abstract

Consider a graph with a path of order . What conditions force 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 as a subgraph. In this case we show that has an induced path of order . This is an exponential improvement over a result of Galvin, Rival, and Sands (1982) and comes close to a recent upper bound of order . Another way to approach this problem is by viewing as an ordered graph (where the vertices are ordered according to their position on the path ). 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 , then this induced path has size at least , and can be chosen to be increasing with respect to .

Paper Structure

This paper contains 10 sections, 37 theorems, 27 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem 1
  4. Coloring the triples
  5. Concluding the proof
  6. Forbidden patterns
  7. Preliminaries
  8. Matchings
  9. General patterns
  10. Open problems

Key Result

Theorem 1.1

For every $t\in \mathbb{N}$ there is an unbounded function $f\colon \mathbb{N}\to\mathbb{N}$ such that for every graph $G$ the following holds: if $G$ is $K_{t,t}$-subgraph free and has a path of order $n$, then $G$ has an induced path of order at least $f(n)$.

Figures (8)

  • Figure 1: Currently best known bounds regarding \ref{['question']}. We recall that for hereditary graph classes, excluding a bipartite subgraph is the most general setting where the property of \ref{['question']} holds with $f$ unbounded. In the results above, $k$ is a strict upper-bound on the considered parameter and $c$ depends on the excluded graph. Treewidth and pathwidth are respectively abbreviated tw and pw. Arrows point towards more general concepts.
  • Figure 2: The path $P$ (in bold).
  • Figure 3: Non-adjacencies between the subpaths of $P$ (indicated by dashed lines). This excludes the edges of $P$ connecting each subpath to the next subpath.
  • Figure 4: The ordered graphs $G_2$ (left), and $G_3$ (right), where vertices are ordered from left to right.
  • Figure 5: The ordered matchings $H$ and $\widehat{H}$ appearing in the proof of Theorem \ref{['thm:planar']}.
  • ...and 3 more figures

Theorems & Definitions (64)

  • Theorem 1.1: galvin1982ramsey
  • Theorem 1.2
  • Corollary 1.3
  • Corollary 1.4
  • Corollary 1.5
  • Theorem 1.6: II
  • Theorem 2.1: erdos1952combinatorial
  • Lemma 3.1
  • proof
  • Remark 3.2
  • ...and 54 more