5 research outputs found
Counting cycles in planar triangulations
We investigate the minimum number of cycles of specified lengths in planar
-vertex triangulations . It is proven that this number is for
any cycle length at most , where denotes the radius
of the triangulation's dual, which is at least logarithmic but can be linear in
the order of the triangulation. We also show that there exist planar
hamiltonian -vertex triangulations containing many -cycles for any
. Furthermore, we prove
that planar 4-connected -vertex triangulations contain many
-cycles for every , and that, under certain
additional conditions, they contain -cycles for many values of
, including
Circumference of 3-connected claw-free graphs and large Eulerian subgraphs of 3-edge-connected graphs
AbstractThe circumference of a graph is the length of its longest cycles. Results of Jackson, and Jackson and Wormald, imply that the circumference of a 3-connected cubic n-vertex graph is Ω(n0.694), and the circumference of a 3-connected claw-free graph is Ω(n0.121). We generalize and improve the first result by showing that every 3-edge-connected graph with m edges has an Eulerian subgraph with Ω(m0.753) edges. We use this result together with the Ryjáček closure operation to improve the lower bound on the circumference of a 3-connected claw-free graph to Ω(n0.753). Our proofs imply polynomial time algorithms for finding large Eulerian subgraphs of 3-edge-connected graphs and long cycles in 3-connected claw-free graphs
The circumference of a graph with no K3,t-minor, II
The class of graphs with no K3;t-minors, t>=3, contains all planar graphs and plays an important role in graph minor theory. In 1992, Seymour and Thomas conjectured the existence of a function α(t)>0 and a constant β>0, such that every 3-connected n-vertex graph with no K3;t-minors, t>=3, contains a cycle of length at least α(t)nβ. The purpose of this paper is to con¯rm this conjecture with α(t)=(1/2)t(t-1) and β=log1729 2.preprin
Long cycles in 3-connected graphs in orientable surfaces
In this paper we apply a cutting theorem of Thomassen to show that there is a function f: N → N such that if G is a 3-connected graph which can be embedded in the orientable surface of genus g with face-width at least f(g), then G contains a cycle of length at least c(n log 3 2), where c is a constant not dependent on g. ∗ MSC Primary 05C38 and 05C50 Secondary 57M1