A quasi-optimal upper bound for induced paths in sparse graphs

In 2012, Nešetřil and Ossona de Mendez proved that graphs of bounded degeneracy that have a path of order $n$ also have an induced path of order $Ω(\log \log n)$. In this paper we give an almost matching upper bound by describing, for arbitrarily large values of $n$, 2-degenerate graphs that have a path of order $n$ and where the longest induced paths have order $O((\log \log n)^{1+o(1)})$.

💡 Research Summary

The paper investigates the relationship between ordinary paths and induced paths in sparse graphs, focusing on k‑degenerate graphs. In 2012, Nešetřil and Ossona de Mendez proved that any k‑degenerate graph containing a path of order n must contain an induced path of length at least Ω(log log n / log(k+1)). The natural question, posed in the same work, is whether this lower bound is tight. Prior to this paper, the best known upper bound for the case k = 2 was O((log log n)²), obtained by the second and third authors in 2023.

The main contribution is Theorem 1.5, which asserts that for infinitely many integers n there exists a 2‑degenerate graph G with a Hamiltonian path of length n, yet every induced path in G has length at most c·log log n·log log log n for some absolute constant c. This improves the previous O((log log n)²) bound by a factor of log log n and matches the known lower bound up to a log log log n factor, thereby essentially settling the optimal order of magnitude for f_k(n) and h_t(n) (the functions governing the guaranteed induced‑path length in k‑degenerate and K_{t,t}‑free graphs, respectively).

The construction proceeds in three major stages:

1. Skeleton Tree (S_ℓ). For a parameter ℓ (chosen so that 2^{2ℓ} ≈ n), the authors start with a complete binary tree S_ℓ of depth 2ℓ − 1, which is trivially 2‑degenerate. However, S_ℓ lacks a Hamiltonian path and its longest induced paths are too long (Θ(2^ℓ)).

2. Ribbed Tree via Barriers. An auxiliary ℓ‑index‑tree T (a partial binary tree whose vertices are the integers 1…ℓ) is introduced. For each index i in T a “full index‑barrier” B(i) is defined recursively as a word over the alphabet {1,…,i}. These words encode a pattern of edges that will replace each edge of the skeleton tree. The replacement creates a “ribbed‑tree” R(T) in which every original edge is substituted by a barrier path together with “ribs” that connect the barrier vertices to carefully chosen ancestors in S_ℓ. The barriers are designed so that any induced path that tries to traverse the graph must respect a nesting of “ranks” associated with the vertices; the rank of a vertex corresponds to the index of the barrier it belongs to. As an induced path proceeds, the ranks of visited vertices must become strictly nested within the index‑tree, which forces the path length to be bounded by the size of the largest barrier.

3. Blow‑up to Obtain a Hamiltonian Path. The ribbed‑tree still does not contain a Hamiltonian path. The authors apply a blow‑up operation that replaces each vertex of R(T) by a short path and connects these copies in a linear order, preserving 2‑degeneracy. The resulting graph G(T) has Θ(2^{2ℓ}) vertices, contains a Hamiltonian path, and inherits the barrier‑induced bound on induced‑path length.

The technical heart of the proof lies in the rank‑nesting argument. By assigning each vertex a rank (the index of its barrier) and defining a “zone” that captures the sub‑tree of S_ℓ associated with that rank, the authors show that any induced path cannot revisit a rank once it has moved to a deeper zone. Consequently, the number of vertices that an induced path can visit is at most the total length of the largest barrier. By carefully choosing the index‑tree T, the authors balance the depth of T against the size of its left sub‑trees, achieving barriers of size O(ℓ log ℓ). Since ℓ ≈ log log n, this yields the claimed O(log log n·log log log n) bound.

The paper also derives immediate corollaries for the functions h_t(n) (the guaranteed induced‑path length in K_{t,t}‑free traceable graphs) and f_k(n) (the analogous function for k‑degenerate graphs). For every integer t ≥ 3 and all sufficiently large n, the bounds c_t·log log n·log log log n ≤ h_t(n) ≤ c·log log n·log log log n hold, and for every k ≥ 1, log log n·log(k+1) ≤ f_k(n) ≤ c·log log n·log log log n. Thus the paper essentially settles the asymptotic behavior of these functions up to a log log log n factor.

Beyond the combinatorial interest, the results have algorithmic implications. Improved bounds on h_t(n) translate into tighter parameterized complexity estimates for the Biclique problem, and the relationship with treedepth suggests structural consequences for sparse graph classes. The construction also enriches the toolbox for building extremal examples in degeneracy‑based graph theory. Overall, the work delivers a near‑optimal upper bound for induced paths in sparse graphs, closing a long‑standing gap between lower and upper bounds.