On planar 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 v≠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 Thomassen's question whether 4‐connected hypohamiltonian graphs exist remains open. We construct planar almost hypohamiltonian graphs of order n for every n≥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 Chvátal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems.

“…This shows that, restricted to this particular family, the result of Araya and Wiener published in [141] is bestpossible.…”

Section: Lemma 23 [80]mentioning

confidence: 67%

Section: Generation Of 4-face Deflatable Hypohamiltonian Graphsmentioning

confidence: 74%

Section: The Planar Casementioning

confidence: 95%

“…Theorem 2.9 (Araya and Wiener, 2011 [141]) There exists a planar hypohamiltonian graph on n vertices for every n ≥ 76.…”

Section: Possible Orders Of (Planar) Hypohamiltonian Graphsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

On Hypohamiltonian and Almost Hypohamiltonian Graphs

Zamfirescu

2014

Planar Hypohamiltonian Graphs on 40 Vertices

Jooyandeh

McKay

Östergård

et al. 2016

A graph is hypohamiltonian if it is not Hamiltonian, but the deletion of any single vertex gives a Hamiltonian graph. Until now, the smallest known planar hypohamiltonian graph had 42 vertices, a result due to Araya and Wiener. That result is here improved upon by 25 planar hypohamiltonian graphs of order 40, which are found through computeraided generation of certain families of planar graphs with girth 4 and a fixed number of 4-faces. It is further shown that planar hypohamiltonian graphs exist for all orders greater than or equal to 42. If Hamiltonian cycles are

Leaf‐Critical and Leaf‐Stable Graphs

Wiener

2016

Self Cite

Abstract:The minimum leaf number ml(G) of a connected graph G is defined as the minimum number of leaves of the spanning trees of G if G is not hamiltonian and 1 if G is hamiltonian. We study nonhamiltonian graphs with the property l :. These graphs will be called (l + 1)-leaf-critical and l -leaf-stable, respectively. It is far from obvious whether such graphs exist; for example, the existence of 3-leaf-critical graphs (that turn out to be the so-called hypotraceable graphs) was an open problem until 1975. We show that l -leaf-stable and l -leaf-critical graphs exist for every integer l ≥ 2, moreover for n sufficiently large, planar l -leafstable and l -leaf-critical graphs exist on n vertices. We also characterize 2-fragments of leaf-critical graphs generalizing a lemma of Thomassen. As an application of some of the leaf-critical graphs constructed, we settle an open problem of Gargano et al. concerning spanning spiders. We also explore connections with a family of graphs introduced by Grünbaum in correspondence with the problem of finding graphs without concurrent longest paths.