Structural and computational results on platypus graphs

TL;DR

This paper explores the properties, existence, and computational classification of platypus graphs, a special class of non-hamiltonian graphs with traceable vertex-deleted subgraphs, including new results on girth, degree, and exhaustive enumeration.

Contribution

It provides the first comprehensive analysis of cubic platypus graphs across all orders, constructs new examples with high girth, and computationally classifies all non-isomorphic platypus graphs for various parameters.

Findings

Complete classification of platypus graphs for certain orders and girths.

Existence of non-cubic platypus graphs with girth up to 16.

New theoretical bounds on maximum degree in platypus graphs.

Abstract

A platypus graph is a non-hamiltonian graph for which every vertex-deleted subgraph is traceable. They are closely related to families of graphs satisfying interesting conditions regarding longest paths and longest cycles, for instance hypohamiltonian, leaf-stable, and maximally non-hamiltonian graphs. In this paper, we first investigate cubic platypus graphs, covering all orders for which such graphs exist: in the general and polyhedral case as well as for snarks. We then present (not necessarily cubic) platypus graphs of girth up to 16---whereas no hypohamiltonian graphs of girth greater than 7 are known---and study their maximum degree, generalising two theorems of Chartrand, Gould, and Kapoor. Using computational methods, we determine the complete list of all non-isomorphic platypus graphs for various orders and girths. Finally, we address two questions raised by the third author in [J. Graph Theory \textbf{86} (2017) 223--243].