Planar Hypohamiltonian Graphs on 40 Vertices

Abstract

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

Seven Problems on Hypohamiltonian and Almost Hypohamiltonian Graphs

“…In [4], together with Jooyandeh, McKay,Östergård, and Pettersson, we showed that there exist 25 planar hypohamiltonian graphs of order 40. Despite the progress made in [4], there still is a considerable gap between the order of the smallest known planar hypohamiltonian graph, which is 40, and the best lower bound known for the order of the smallest such graphs, which is 18, see [1]. (a) Do planar hypohamiltonian graphs on less than 40 vertices exist?…”

On constructions of hypotraceable graphs

“…Until now all hypotraceable graphs were constructed using hypohamiltonian graphs; extending a method of Thomassen [8] we present a construction that uses so-called almost hypohamiltonian graphs (nonhamiltonian graphs, whose vertex deleted subgraphs are hamiltonian with exactly one exception). As an application, we construct a planar hypotraceable graph of order 138, improving the best known bound of 154 [5]. We also prove a structural type theorem showing that hypotraceable graphs possessing some connectivity properties are all built using either Thomassen's or our method.…”

“…It is worth mentioning that until 1976 all hypohamiltonian graphs known were non-planar (actually, of crossing number at least 2), which led Chvátal to ask whether planar hypohamiltonian graphs exist [1]. The first such graph (of order 105) was found by Thomassen [9], the smallest known planar hypohamiltonian, hypotraceable, and almost hypohamiltonian graphs have order 40 [5], 154 [5], and 39 [11], respectively. For more about hypohamiltonicity and hypotraceablity, see the survey paper by Holton and Sheehan [4] The paper is organized as follows.…”

“…As it was mentioned in the Introduction, the smallest planar hypotraceable graph (which is based on four copies of the 40 vertex planar hypohamiltonian graph of Jooyandeh et al [5]) is of order 154, and appears also in [5]. In order to improve on this, we present a 36 vertex planar almost hypohamiltonian graph G (Figure 1) with exceptional vertex w of degree 3, such that the neighbours of w also have degree 3.…”

Forbidden configurations for hypohamiltonian graphs

“…The smallest such graph is the Petersen graph (see [8]) and, by results from [2][3][4]11], and [1], n-vertex hypohamiltonian graphs exist for all n ≥ 10, n ∈ {11, 12, 14, 17} (even more, the number of nonisomorphic such graphs grows exponentially with n, see [10]). A lot of work was also done in study of planar hypohamiltonian graphs, concerning mainly their constructions and looking for smallest examples, see [12,7,17,15,9]. In this paper, we are interested in forbidden configurations for hypohamiltonian graphs.…”