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

Carsten Thomassen

CT Zamfirescu

REL Aldred

English

Zamfirescu, Carol

Ghent University Academic Bibliography

Seven problems on hypohamiltonian andalmost hypohamiltonian graphsCarol T. Zamrescu2010 Mathematics Subject Classication: 05C10, 05C38, 05C45.Here, all graphs are undirected, nite, connected, and neither contain loops nor multipleedges. Let G be a graph. If a vertex of G has degree 3, then we call the vertex cubic. Ifevery vertex of G is cubic, we call G cubic. G is hamiltonian if it contains a hamiltoniancycle, i.e. a cycle passing through all vertices of the graph. G is hypohamiltonian if Gdoes not contain a hamiltonian cycle but for any vertex v of G, the graph G   v doescontain a hamiltonian cycle. If we replace \cycle" by \path", we obtain the denition ofa hypotraceable graph.Let n0 (c0) be the smallest natural number such that there exists a planar (planarcubic) hypohamiltonian graph of order n for every n  n0 (for every even n  c0).The solution to the rst problem we want to present would constitute an improvementto the main result from [4]. We consider it to be very interesting, since many consequencesconcerning problems on longest paths and cycles arise, but also very dicult. Similarly,we consider Problem 4 to be especially intriguing.1. In [4], together with Jooyandeh, McKay, Ostergard, and Pettersson, we showed thatthere exist 25 planar hypohamiltonian graphs of order 40. Despite the progress madein [4], there still is a considerable gap between the order of the smallest known planarhypohamiltonian graph, which is 40, and the best lower bound known for the order of thesmallest such graphs, which is 18, see [1].(a) Do planar hypohamiltonian graphs on less than 40 vertices exist?(b) What is n0? (It is known that n0  42, see [4].)The author is a PhD fellow at Ghent University on the BOF (Special Research Fund) scholarship01DI1015.12. Araya and Wiener showed that there exists a planar cubic hypohamiltonian graph on70 vertices [2].(a) Is there such a graph with fewer vertices?(b) Correspondingly, we ask for c0. (Presently, we know that c0  74, see [6].)A graph G is almost hypohamiltonian if G is non-hamiltonian, and there exists a vertexw, which we will call exceptional, such that G w is non-hamiltonian, yet for any vertexv 6= w the graph G  v is hamiltonian. Dene n00 to be the smallest natural number suchthat there exists a planar almost hypohamiltonian graph of order n for every n  n00.3. (a) We know that there exists an almost hypohamiltonian graph on 17 vertices [6].Are there smaller examples?(b) Do almost hypohamiltonian graphs of order n 2 f18; 19; 21; 24g exist?4. Thomassen's question from 1978 whether 4-connected hypohamiltonian graphs existremains open [5]. We ask here:(a) Do 4-connected hypotraceable graphs exist? (Horton [3] showed that 3-connectedhypotraceable graphs exist.)(b) Do 5-connected almost hypohamiltonian graphs exist? (We know that 4-connectedalmost hamiltonian graphs do exist [6].)Note that, by results from [6], a positive answer to (a) would imply a positive answerto (b), but not necessarily vice-versa.5. Is there an almost hypohamiltonian graph with a cubic exceptional vertex and allother vertices of degree at least 4? Solving this would answer, by using a result from [6],Thomassen's question whether hypohamiltonian graphs with minimum degree 4 exist [5].6. (a) Is there a planar almost hypohamiltonian graph with fewer than 39 vertices? (Anexample with 39 vertices is available [6].)(b) What is n00? (So far, it is known that n00  76, see [6].)7. Due to certain gluing results, planar almost hypohamiltonian graphs with cubic excep-tional vertex are of special interest. In [6] it is shown that there exists a planar almosthypohamiltonian graph of order 47 whose exceptional vertex is cubic. Are there smallerexamples? (This would help improve the bound for n00.)2References[1] R. E. L. Aldred, B. D. McKay, and N. C. Wormald. Small Hypohamiltonian Graphs.J. Combin. Math. Combin. Comput. 23 (1997) 143{152.[2] M. Araya and G. Wiener. On cubic planar hypohamiltonian and hypotraceablegraphs. Electron. J. Combin. 18 (2011) #P85.[3] J. D. Horton. A Hypotraceable Graph. Research Report CORR 73{4, Dept. Combin.and Optim., Univ. Waterloo, 1973.[4] M. Jooyandeh, B. D. McKay, P. R. J. Ostergard, V. H. Pettersson, andC. T. Zamrescu. Planar hypohamiltonian graphs on 40 vertices. Submitted.[5] C. Thomassen. Hypohamiltonian graphs and digraphs. Theory and Applications ofGraphs, Lecture Notes in Mathematics 642, Springer, Berlin (1978) 557{571.[6] C. T. Zamrescu. On hypohamiltonian and almost hypohamiltonian graphs. J. GraphTheory 79 (2015) 63{81.Carol T. ZamrescuDepartment of Applied Mathematics, Computer Science and Statistics,Ghent University, Krijgslaan 281 - S9, 9000 Ghent, Belgium.czamrescu@gmail.com3

Seven problems on hypohamiltonian and almost 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 vw 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 n74. 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 Chvatal (originally disproved by Thomassen), strengthen a result of Araya and Wiener on cubic planar hypohamiltonian graphs, and mention open problems

Hypohamiltonian and almost hypohamiltonian graphs

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Ghent University Academic Bibliography