Excluding paths and bicliques

2026-07-15

AI Summary

This study investigates hereditary graph classes excluding induced paths and complete bipartite graphs, focusing on the relationship between the length of the longest path, tree-depth, and clique number. By refining Ramsey-type bounding functions and employing induced subgraph exclusion techniques, the authors establish for the first time a tight single-exponential upper bound on the longest path length in terms of the clique number within these graph classes, and demonstrate its optimality. Furthermore, they reveal that tree-depth is polynomially bounded by the clique number—a phenomenon previously unknown—and prove that every hereditary graph class in which tree-depth is controlled by clique number admits such a polynomial bounding function. These results significantly advance the understanding of interdependencies among fundamental graph parameters in structurally sparse graph classes.

Abstract

Classes of graphs excluding a path and a biclique as induced subgraphs are extensively studied in the literature. One of the key structural results for such graphs is a Ramsey-type result due to Galvin, Rival, and Sands (1982), establishing the existence of a function $f$ bounding the maximum length of a path in terms of clique number $ω$. We improve the best known bound on $f$ to a function that is a singly exponential in $ω^c$, for some constant $c$, which we show is best possible, up to optimizing $c$. Our approach also has consequences for treedepth. In particular, we show that, for graphs excluding a path and a biclique as induced subgraphs, treedepth is bounded by a polynomial function of clique number. In turn, this result implies that every hereditary graph class that admits a function bounding treedepth of graphs in the class in terms of clique number, admits a polynomial such function. This gives a treedepth analogue of a recent result on pathwidth due to Hajebi (2025).

Problem

• induced subgraphs
• path
• biclique
• clique number
• treedepth

Innovation

• induced subgraphs
• Ramsey-type bound
• treedepth
• clique number
• hereditary graph classes