Long induced paths in expanders

TL;DR

The paper addresses finding long induced paths in sparse graphs, extending known results from random graphs to spectral expanders. It introduces a constructive DFS-like algorithm that maintains a path induced in the host graph while controlling edge-density via two global bounds, enabling the extraction of long induced paths in time $O(e(G))$. The main contributions are (i) a linear-length induced path result for $(n,d,\lambda)$-graphs with $\lambda<d^{3/4}/100$ and $d<n/10$, (ii) generalization to graphs satisfying a mild upper-uniformity condition, and (iii) applications to induced size-Ramsey numbers of paths, including a multicolor bound $\hat{r}_{ind}^k(P_n)=O(k^3\log^2 k)\,n$ and a random-graph corollary for $G(nk,p)$ with $p=(c\log k)/n$. The approach provides a constructive, algorithmic perspective on long induced substructures in expanders and sparse graphs, connecting deterministic spectral properties with Ramsey-type phenomena and probabilistic models.

Abstract

We prove that any bounded degree regular graph with sufficiently strong spectral expansion contains an induced path of linear length. This is the first such result for expanders, strengthening an analogous result in the random setting by Draganić, Glock and Krivelevich. More generally, we find long induced paths in sparse graphs that satisfy a mild upper-uniformity edge-distribution condition.

Figures (1)

• Figure 1: The algorithm used in the proof of \ref{['thm:DFSnew']}.

Theorems & Definitions (5)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 3.1
• Corollary 3.2