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

TL;DR

The paper tackles the growth of induced paths in sparse graphs, focusing on 2-degenerate graphs with Hamiltonian paths. It introduces a barrier-based construction—comprising skeleton-trees, index-trees, ribbed-trees, and a blow-up operation—to produce large 2-degenerate graphs in which induced paths remain short. The authors prove that for infinitely many n there exist 2-degenerate graphs with a path of length n and no induced path longer than $c \cdot \log \log n \cdot \log \log \log n$, achieving a quasi-optimal upper bound up to a triple-log factor. This advances the understanding of induced-path bounds in sparse graph classes and tightens the gap with known lower bounds, with potential implications for related parameters such as treedepth and algorithmic problems tied to induced substructures.

Abstract

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 $O((\log \log n)^{1+o(1)})$.

Figures (13)

• Figure 1: The different steps in the construction of the graph satisfying Theorem \ref{['thm:main-theorem']}, detailed in Section \ref{['sec:construction']}, starting from an arbitrary integer $\ell$.
• Figure 2: Inductive construction of a skeleton tree (${\sf S}_2$ on the left, ${\sf S}_\ell$ on the right) and related terminology. Dotted lines denote the ranks of the nodes lying at the same depth in the tree. Zones are depicted in shaded areas.
• Figure 3: A 7-index-tree ${\sf T}$, its associated full index-barrier $B(7)$, and two index-barriers $B(6,1)$ and $B(4,3)$. For better readability, to each index is associated a distinct color. Recall that index-trees are not necessarily complete binary trees.
• Figure 4: The construction of the barriers in the two cases where $4=i\geq 2$ (left) and $i=1$ (right), for $\ell=7$ and with ${\sf T}$ being a complete binary tree as in Figure \ref{['fig:index-tree']}. The ranks of the nodes are indicated by colored integers. The shaded areas correspond to the zones, with the zone of $6$ partially represented on the left for better readability: it should be understood that it contains the two other represented zones as well. The square node (right) represents the extra subdivision $z$ in the case where $i=1$. Note that the obtained subdivided nodes are considered in the zone of their parent node $s$.
• Figure 5: The blowup of a ribbed-tree detailed on the two cases where $i\geq 2$ (actually $i=4$) on the left and $i=1$ on the right, for $\ell=7$. The groups of 3 consecutive vertices $x_1,x_2,x_3$ representing an index of the index-barrier are represented by rectangles, and detailed only once in the right barrier of the case $i\geq 2$ (left). Spiky rectangles represent vertices incident to hopping ribs. Thick edges represent the top edges of the triangles.

Theorems & Definitions (53)

• Theorem 1.1: see ding1996unavoidable
• Theorem 1.2: galvin1982ramsey
• Theorem 1.3: nevsetvril2012sparsity
• Theorem 1.5
• Remark 3.1
• Remark 3.2
• Definition 3.3
• Proposition 3.7
• proof
• Proposition 3.8