Carol T. Zamfirescu

On Hypohamiltonian and Almost Hypohamiltonian Graphs

A graph G is almost hypohamiltonian if G is non-hamiltonian, there exists a vertex w such that G-w is non-hamiltonian, and for any vertex vi¾?w the graph G-v is hamiltonian. We prove the existence of an almost hypohamiltonian graph with 17 vertices and of a planar such graph with 39 vertices. Moreover, we find a 4-connected almost hypohamiltonian graph, while Thomassens question whether 4-connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every ni¾?74. During our investigation we draw connections between hypotraceable, hypohamiltonian, and almost hypohamiltonian graphs, and discuss a natural extension of almost hypohamiltonicity. Finally, we give a short argument disproving a conjecture of Chvatal originally disproved by Thomassen, strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.

Survey of two-dimensional acute triangulations

Abstract We give a brief introduction to the topic of two-dimensional acute triangulations, mention results on related areas, survey existing achievements–with emphasis on recent activity–and list related open problems, both concrete and conceptual.

On hypohamiltonian snarks and a theorem of Fiorini

We discuss an omission in the statement and proof of Fiorinis 1983 theorem on hypohamiltonian snarks and present a version of this theorem which is more general in several ways. Using Fiorinis erroneous result, Steffen showed that hypohamiltonian snarks exist for some

Lattice graphs with non-concurrent longest cycles

n \ge 10

Every graph occurs as an induced subgraph of some hypohamiltonian graph

and each even

An Infinite Family of Planar Non-Hamiltonian Bihomogeneously Traceable Oriented Graphs

n \ge 92

Grinberg’s Criterion

. We rectify Steffens proof by providing a correct demonstration of a technical lemma on flower snarks, which might be of separate interest. We then strengthen Steffens theorem to the strongest possible form by determining all orders for which hypohamiltonian snarks exists. This also strengthens a result of Macajova and Skoviera. Finally, we verify a conjecture of Steffen on hypohamiltonian snarks up to 36 vertices.

Cubic vertices in planar hypohamiltonian graphs: ZAMFIRESCU

No hypohamiltonian graphs are embeddable in the planar square lattice. This lattice contains, however, graphs in which every vertex is missed by some longest cycle. In this paper we present graphs with this property, embeddable in various lattices, and of remarkably small order.

Infinitely many planar cubic hypohamiltonian graphs of girth 5

We prove the titular statement. This settles a problem of Chvatal from 1973 and encompasses earlier results of Thomassen, who showed it for K_3, and Collier and Schmeichel, who proved it for bipartite graphs. We also show that for every outerplanar graph there exists a planar hypohamiltonian graph containing it as an induced subgraph.

Hamiltonian properties of polyhedra with few 3-cuts : a survey

We answer an open question on planar non-hamiltonian bihomogeneously traceable digraphs without opposite arcs by constructing an infinite family of such graphs.