6 research outputs found
Generation of cubic graphs and snarks with large girth
We describe two new algorithms for the generation of all non-isomorphic cubic
graphs with girth at least which are very efficient for
and show how these algorithms can be efficiently restricted to generate snarks
with girth at least .
Our implementation of these algorithms is more than 30, respectively 40 times
faster than the previously fastest generator for cubic graphs with girth at
least 6 and 7, respectively.
Using these generators we have also generated all non-isomorphic snarks with
girth at least 6 up to 38 vertices and show that there are no snarks with girth
at least 7 up to 42 vertices. We present and analyse the new list of snarks
with girth 6.Comment: 27 pages (including appendix
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) 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
Graphs with few Hamiltonian Cycles
We describe an algorithm for the exhaustive generation of non-isomorphic
graphs with a given number of hamiltonian cycles, which is especially
efficient for small . Our main findings, combining applications of this
algorithm and existing algorithms with new theoretical results, revolve around
graphs containing exactly one hamiltonian cycle (1H) or exactly three
hamiltonian cycles (3H). Motivated by a classic result of Smith and recent work
of Royle, we show that there exist nearly cubic 1H graphs of order iff is even. This gives the strongest form of a theorem of Entringer and
Swart, and sheds light on a question of Fleischner originally settled by
Seamone. We prove equivalent formulations of the conjecture of Bondy and
Jackson that every planar 1H graph contains two vertices of degree 2, verify it
up to order 16, and show that its toric analogue does not hold. We treat
Thomassen's conjecture that every hamiltonian graph of minimum degree at least
contains an edge such that both its removal and its contraction yield
hamiltonian graphs. We also verify up to order 21 the conjecture of Sheehan
that there is no 4-regular 1H graph. Extending work of Schwenk, we describe all
orders for which cubic 3H triangle-free graphs exist. We verify up to order
Cantoni's conjecture that every planar cubic 3H graph contains a triangle,
and show that there exist infinitely many planar cyclically 4-edge-connected
cubic graphs with exactly four hamiltonian cycles, thereby answering a question
of Chia and Thomassen. Finally, complementing work of Sheehan on 1H graphs of
maximum size, we determine the maximum size of graphs containing exactly one
hamiltonian path and give, for every order , the exact number of such graphs
on vertices and of maximum size.Comment: 29 pages; to appear in Mathematics of Computatio