Higher-dimensional counterexamples to Hamiltonicity

TL;DR

This paper demonstrates that for dimensions other than three, all graphs of d-polytopes have Hamiltonian line graphs, but provides counterexamples in three dimensions and simplicial complexes, challenging existing assumptions in Hamiltonian graph theory.

Contribution

It introduces higher-dimensional counterexamples to Hamiltonicity in polytope line graphs and extends known results to simplicial complexes, revealing limitations of previous theorems.

Findings

Existence of 3-polytope graphs without Hamiltonian paths

Construction of large 3-polytope line graphs with short paths

Elementary counterexamples to Hamiltonian extensions in simplicial complexes

Abstract

For $d \ge 2$, we show that all graphs of $d$-polytopes have a Hamiltonian line graph if and only if $d \ne 3$: We exhibit a graph of a $3$-polytope on $252$ vertices whose line graph does not even have Hamiltonian paths. Adapting a construction by Gr\"unbaum and Motzkin, for large $n$ we also construct simple $3$-polytopes on $3n$ vertices in whose line graph any simple path is shorter than $10 n^{\alpha}$, for some constant $\alpha<1$. Moreover, we give four elementary counterexamples of plausible extensions to simplicial complexes of four famous results in Hamiltonian graph theory.