4 research outputs found
Exact bipartite Tur\'an numbers of large even cycles
Let the bipartite Tur\'an number of a graph be the maximum
number of edges in an -free bipartite graph with two parts of sizes and
, respectively. In this paper, we prove that
for any positive integers with . This
confirms the rest of a conjecture of Gy\"{o}ri \cite{G97} (in a stronger form),
and improves the upper bound of obtained by Jiang and Ma
\cite{JM18} for this range. We also prove a tight edge condition for
consecutive even cycles in bipartite graphs, which settles a conjecture in
\cite{A09}. As a main tool, for a longest cycle in a bipartite graph, we
obtain an estimate on the upper bound of the number of edges which are incident
to at most one vertex in . Our two results generalize or sharpen a classical
theorem due to Jackson \cite{J85} in different ways.Comment: Revised version; 16 page
Cycle spectra of Hamiltonian graphs
AbstractWe prove that every Hamiltonian graph with n vertices and m edges has cycles with more than pβ12lnpβ1 different lengths, where p=mβn. For general m and n, there exist such graphs having at most 2βp+1β different cycle lengths