13 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
Every graph occurs as an induced subgraph of some hypohamiltonian graph
We prove the titular statement. This settles a problem of Chvátal 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 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) 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, Östergård, Pettersson, and the second author
Generation and New Infinite Families of -hypohamiltonian Graphs
We present an algorithm which can generate all pairwise non-isomorphic
-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal
of any pair of adjacent vertices yields a hamiltonian graph, of a given order.
We introduce new bounding criteria specifically designed for
-hypohamiltonian graphs, allowing us to improve upon earlier computational
results. Specifically, we characterise the orders for which
-hypohamiltonian graphs exist and improve existing lower bounds on the
orders of the smallest planar and the smallest bipartite -hypohamiltonian
graphs. Furthermore, we describe a new operation for creating
-hypohamiltonian graphs that preserves planarity under certain conditions
and use it to prove the existence of a planar -hypohamiltonian graph of
order for every integer . Additionally, motivated by a theorem
of Thomassen on hypohamiltonian graphs, we show the existence
-hypohamiltonian graphs with large maximum degree and size.Comment: 21 page
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
On Fulkerson conjecture
If is a bridgeless cubic graph, Fulkerson conjectured that we can find 6
perfect matchings (a{\em Fulkerson covering}) with the property that every edge
of is contained in exactly two of them. A consequence of the Fulkerson
conjecture would be that every bridgeless cubic graph has 3 perfect matchings
with empty intersection (this problem is known as the Fan Raspaud Conjecture).
A {\em FR-triple} is a set of 3 such perfect matchings. We show here how to
derive a Fulkerson covering from two FR-triples. Moreover, we give a simple
proof that the Fulkerson conjecture holds true for some classes of well known
snarks.Comment: Accepted for publication in Discussiones Mathematicae Graph Theory;
Discussiones Mathematicae Graph Theory (2010) xxx-yy