Nonhamiltonian 3-Connected Cubic Planar Graphs

Abstract

Abstract. 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 examples 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 vertices. The fact that all 3-connected cubic planar graphs on at most 176 vertices and with face size at most 6 are hamiltonian is also verified.Key words. nonhamiltonian, cubic, planar AMS subject classification. 05C38 PII. S0895480198348665 1. Introduction. In this paper we describe an investigation (making much use of computation) of cyclically k-connected cubic planar graphs (CkCP s) for k = 4, 5 and report the results. We shall also have occasion to consider cubic 3-connected planar graphs with no restriction on cyclic connectivity; these we refer to as C3CP s. The investigation extends the work of Holton and McKay in [10], in which the smallest order of a nonhamiltonian C3CP was shown to be 38. We provide answers to two questions raised in that paper, namely, the following:(a) What is the smallest order of a nonhamiltonian C4CP? (b) Is there more than one nonhamiltonian C5CP on 44 vertices? The answer to the former question is determined to be 42, and all nonhamiltonian C4CPs of that order are presented. The latter question is answered in the negative, and a complete list of all nonhamiltonian C5CPs on at most 52 vertices is presented.Before proceeding, we include some definitions and results from the existing literature as we shall make use of them in the rest of the paper. By a k-gon we mean a face of a plane graph bounded by k edges. Note that a k-cycle is not necessarily a k-gon. By a k-cut we mean a set of k edges whose removal leaves the graph disconnected and of which no proper subset has that property. The two components (and clearly there are only two) formed by the removal of a k-cut are called k-pieces. A k-cut is nontrivial if each of its k-pieces contains a cycle and is essential if it is nontrivial and each of its k-pieces contains more than k vertices. A cubic graph is cyclically k-connected if it has no nontrivial t-cuts for 0 ≤ t ≤ k − 1, and it has cyclic connectivity k if, in addition, it has at least one nontrivial k-cut. We denote by λ (G) the value of k such that the C3CP G has cyclic connectivity k.If G is a hamiltonian C3CP, then an a-edge is an edge which is present in every hamiltonian cycle in G, while a b-edge is absent from every hamiltonian cycle in G.To focus our search for nonhamiltonian C4CPs of smallest order we shall make use of the following theorem which formed the main result in [10].

On cyclic edge-connectivity of fullerenes

“…This conjecture has been verified for graphs on at most 176 vertices [1]. However, the most general result on this topic is that leapfrog-fullerenes, that is, fullerenes obtained by performing the so called tripling (leapfrog transformation) [8,32], of order congruent to 2 modulo 4 have a Hamilton cycle and that leapfrog-fullerenes of order divisible by 4 have a Hamilton path (see [23]).…”

Cubic vertices in planar hypohamiltonian graphs

“…In case 1 2 3 4 , and once more we have a contradiction. □ Let n be the family of all planar hypohamiltonian graphs of order n, and V G ( ) 3 the set of all cubic vertices in a graph G.…”

“…For results treating the planar case and which are not included in [15], we refer to the works of Aldred, Bau, Holton, and McKay [3], the author and Zamfirescu [35,36], Araya and Wiener [4,33], Jooyandeh, McKay, Östergård, Pettersson, and the author [16], McKay [19], Goedgebeur and the author [11,12], and Wiener [32]. For the situation in directed graphs, see for instance van Aardt et al [1], which answers affirmatively Thomassen's [26,Question 9] from 1976 whether planar hypohamiltonian oriented graphs exist.…”

On Hypohamiltonian and Almost Hypohamiltonian Graphs

“…We have already mentioned that in 1988 it was shown that all planar 3-connected cubic graphs on fewer than 38 vertices are hamiltonian. In 2000, Aldred, Bau, Holton, and McKay [4] proved that there is no planar cubic hypohamiltonian graph on 42 or fewer vertices. They showed that every 3-connected, cyclically 4-edge-connected cubic planar graph has at least 42 vertices and presented all such graphs on exactly 42 vertices.…”

“…In the recent not yet pub- 3 The chromatic index of a graph is the smallest number of colours necessary to colour the edges of the graph such that any two edges sharing an end-point do not have the same colour. 4 In a graph G, M ⊂ E(G) is cycle-separating if G − M is disconnected, and at least two of its components contain cycles. (Note that there exist graphs with no cycle-separating sets, for instance K 3,3 .)…”