47 research outputs found
On almost hypohamiltonian graphs
A graph is almost hypohamiltonian (a.h.) if is non-hamiltonian, there
exists a vertex in such that is non-hamiltonian, and is
hamiltonian for every vertex in . The second author asked in [J.
Graph Theory 79 (2015) 63--81] for all orders for which a.h. graphs exist. Here
we solve this problem. To this end, we present a specialised algorithm which
generates complete sets of a.h. graphs for various orders. Furthermore, we show
that the smallest cubic a.h. graphs have order 26. We provide a lower bound for
the order of the smallest planar a.h. graph and improve the upper bound for the
order of the smallest planar a.h. graph containing a cubic vertex. We also
determine the smallest planar a.h. graphs of girth 5, both in the general and
cubic case. Finally, we extend a result of Steffen on snarks and improve two
bounds on longest paths and longest cycles in polyhedral graphs due to
Jooyandeh, McKay, {\"O}sterg{\aa}rd, Pettersson, and the second author.Comment: 18 pages. arXiv admin note: text overlap with arXiv:1602.0717
On almost hypohamiltonian graphs
A graph G is almost hypohamiltonian (a.h.) if G is non-hamiltonian, there exists a vertex w in G such that G - w is non-hamiltonian, and G - v is hamiltonian for every vertex v \ne w in G. The second author asked in [J. Graph Theory 79 (2015) 63â81] for all orders for which a.h. graphs exist. Here we solve this problem. To this end, we present a specialised algorithm which generates complete sets of a.h. graphs for various orders. Furthermore, we show that the smallest cubic a.h. graphs have order 26. We provide a lower bound for the order of the smallest planar a.h. graph and improve the upper bound for the order of the smallest planar a.h. graph containing a cubic vertex. We also determine the smallest planar a.h. graphs of girth 5, both in the general and cubic case. Finally, we extend a result of Steffen on snarks and improve two bounds on longest paths and longest cycles in polyhedral graphs due to Jooyandeh, McKay, ĂstergĂ„rd, Pettersson, and the second author
Hypohamiltonian and almost hypohamiltonian graphs
This Dissertation is structured as follows. In Chapter 1, we give a short historical overview and define fundamental concepts. Chapter 2 contains a clear narrative of the progress made towards finding the smallest planar hypohamiltonian graph, with all of the necessary theoretical tools and techniques--especially Grinberg's Criterion. Consequences of this progress are distributed over all sections and form the leitmotif of this Dissertation. Chapter 2 also treats girth restrictions and hypohamiltonian graphs in the context of crossing numbers. Chapter 3 is a thorough discussion of the newly introduced almost hypohamiltonian graphs and their connection to hypohamiltonian graphs. Once more, the planar case plays an exceptional role. At the end of the chapter, we study almost hypotraceable graphs and Gallai's problem on longest paths. The latter leads to Chapter 4, wherein the connection between hypohamiltonicity and various problems related to longest paths and longest cycles are presented. Chapter 5 introduces and studies non-hamiltonian graphs in which every vertex-deleted subgraph is traceable, a class encompassing hypohamiltonian and hypotraceable graphs. We end with an outlook in Chapter 6, where we present a selection of open problems enriched with comments and partial results
Every graph occurs as an induced subgraph of some hypohamiltonian graph
We prove the titular statement. This settles a problem of ChvĂĄtal 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
On the size of maximally non-hamiltonian digraphs
A graph is called maximally non-hamiltonian if it is non-hamiltonian, yet for any two non-adjacent vertices there exists a hamiltonian path between them. In this paper, we naturally extend the concept to directed graphs and bound their size from below and above. Our results on the lower bound constitute our main contribution, while the upper bound can be obtained using a result of Lewin, but we give here a different proof. We describe digraphs attaining the upper bound, but whether our lower bound can be improved remains open
Planar hypohamiltonian oriented graphs
In 1978 Thomassen asked whether planar hypohamiltonian oriented graphs exist. Infinite families of such graphs have since been described but for infinitely many it remained an open question whether planar hypohamiltonian oriented graphs of order exist. In this paper we develop new methods for constructing hypohamiltonian digraphs, which, combined with efficient graph generation algorithms, enable us to fully characterise the orders for which planar hypohamiltonian oriented graphs exist. Our novel methods also led us to discover the planar hypohamiltonian oriented graph of smallest order and size, as well as infinitely many hypohamiltonian orientations of maximal planar graphs. Furthermore, we answer a question related to a problem of Schiermeyer on vertex degrees in hypohamiltonian oriented graphs, and characterise all the orders for which planar hypotraceable oriented graphs exist.Research Foundation Flanders; VSC(Flemish Supercomputer Center);DSTâNRF Centre of Excellence in Mathematical and Statistical Sciences.http://wileyonlinelibrary.com/journal/jgthj2023Mathematics and Applied Mathematic