The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices

The smallest non-hamiltonian 3-connected cubic planar graphs have 38 vertices

Cycles Through 23 Vertices in 3-Connected Cubic Planar Graphs

We establish that if A is a set of at most 23 vertices in a 3-connected cubic planar graph G, then there is a cycle in G containing A. This result is sharp. AMS Classification: 05C38 Let G be a 3-connected cubic planar graph and let A ⊆ V (G). It was shown in [4] that if |A| ≤ 19 there is a cycle C in G such that A ⊆ V (C). In this paper we show that if |A| ≤ 23, then G contains a cycle through...

Nonhamiltonian 3-Connected Cubic Planar Graphs

We establish that every cyclically 4-connected cubic planar graph of order at most 40 is hamiltonian. Furthermore, this bound is determined to be sharp and we present all nonhamiltonian such graphs of order 42. In addition we list all nonhamiltonian cyclically 5-connected cubic planar graphs of order at most 52 and all nonhamiltonian 3-connected cubic planar graphs of girth 5 on at most 46 vert...

Non-Isomorphic Smallest Maximally Non-Hamiltonian Graphs

A graph G is maximally non-hamiltonian (MNH) if G is not hamiltonian but becomes hamiltonian after adding an arbitrary new edge. Bondy 2] showed that the smallest size (= number of edges) in a MNH graph of order n is at least d 3n 2 e for n 7. The fact that equality may hold there for innnitely many n was suggested by Bollobbs 1]. This was connrmed by Clark, Entringer and Shapiro (see 5, 6]) an...

Cubic planar hamiltonian graphs of various types

Let U be the set of cubic planar hamiltonian graphs, A the set of graphs G in U such thatG− v is hamiltonian for any vertex v of G, B the set of graphs G in U such thatG− e is hamiltonian for any edge e of G, and C the set of graphs G in U such that there is a hamiltonian path between any two different vertices of G. With the inclusion and/or exclusion of the sets A,B, and C, U is divided into ...