Density of Traceable Graphs

TL;DR

This paper establishes tight bounds on the number of edges in traceable graphs across various dense graph classes, revealing different growth rates depending on structural parameters.

Contribution

It provides the first comprehensive bounds on edges in traceable graphs for multiple dense graph classes, linking structural parameters to edge density.

Findings

Quadratic bounds for graphs with bounded neighborhood diversity

n log n bounds for cographs and bounded modular-width graphs

Superlinear bounds for graphs with bounded shrub-depth

Abstract

We establish tight lower and upper bounds on the number of edges in traceable graphs in several classes of dense graphs. A graph is traceable if it has a Hamiltonian path. We show that the bound is: - quadratic for the class of graphs of bounded neighborhood diversity, bounded size of maximum induced matching or bounded cluster vertex deletion number; - n log n for the class of cographs or, more generaly, bounded modular-width, and for the class of bounded distance to cograph; and - sligthly superlinear for the class of bounded shrub-depth.