Dirac's theorem for graphs of bounded bandwidth

TL;DR

This paper establishes an optimal condition linking minimum degree and bandwidth that guarantees the presence of a spanning subdivision of a complete bipartite graph in a graph, with applications to Hamilton paths, cycles, and random geometric graphs.

Contribution

It introduces a new sufficient condition involving bandwidth and minimum degree for graphs to contain specific spanning subdivisions, extending Dirac's theorem.

Findings

Provides an optimal condition relating degree and bandwidth for spanning subdivisions.

Includes a greedy algorithm for constructing these structures.

Applications demonstrated in Hamilton paths, cycles, and random geometric graphs.

Abstract

We provide an optimal sufficient condition, relating minimum degree and bandwidth, for a graph to contain a spanning subdivision of the complete bipartite graph $K_{2,\ell}$. This includes the containment of Hamilton paths and cycles, and has applications in the random geometric graph model. Our proof provides a greedy algorithm for constructing such structures.