3 research outputs found
Hamiltonicity, independence number, and pancyclicity
A graph on n vertices is called pancyclic if it contains a cycle of length l
for all 3 \le l \le n. In 1972, Erdos proved that if G is a Hamiltonian graph
on n > 4k^4 vertices with independence number k, then G is pancyclic. He then
suggested that n = \Omega(k^2) should already be enough to guarantee
pancyclicity. Improving on his and some other later results, we prove that
there exists a constant c such that n > ck^{7/3} suffices
Pancyclicity of Hamiltonian and highly connected graphs
A graph G on n vertices is Hamiltonian if it contains a cycle of length n and
pancyclic if it contains cycles of length for all .
Write for the independence number of , i.e. the size of the
largest subset of the vertex set that does not contain an edge, and
for the (vertex) connectivity, i.e. the size of the smallest subset of the
vertex set that can be deleted to obtain a disconnected graph. A celebrated
theorem of Chv\'atal and Erd\H{o}s says that is Hamiltonian if . Moreover, Bondy suggested that almost any non-trivial
conditions for Hamiltonicity of a graph should also imply pancyclicity.
Motivated by this, we prove that if then G is
pancyclic. This establishes a conjecture of Jackson and Ordaz up to a constant
factor. Moreover, we obtain the more general result that if G is Hamiltonian
with minimum degree then G is pancyclic. Improving
an old result of Erd\H{o}s, we also show that G is pancyclic if it is
Hamiltonian and . Our arguments use the following theorem
of independent interest on cycle lengths in graphs: if then G contains a cycle of length for all .Comment: 15 pages, 1 figur