Author:
Draganić Nemanja,Munhá Correia David,Sudakov Benny
Abstract
Abstract
The bipartite independence number of a graph
$G$
, denoted as
$\tilde \alpha (G)$
, is the minimal number
$k$
such that there exist positive integers
$a$
and
$b$
with
$a+b=k+1$
with the property that for any two disjoint sets
$A,B\subseteq V(G)$
with
$|A|=a$
and
$|B|=b$
, there is an edge between
$A$
and
$B$
. McDiarmid and Yolov showed that if
$\delta (G)\geq \tilde \alpha (G)$
then
$G$
is Hamiltonian, extending the famous theorem of Dirac which states that if
$\delta (G)\geq |G|/2$
then
$G$
is Hamiltonian. In 1973, Bondy showed that, unless
$G$
is a complete bipartite graph, Dirac’s Hamiltonicity condition also implies pancyclicity, i.e., existence of cycles of all the lengths from
$3$
up to
$n$
. In this paper, we show that
$\delta (G)\geq \tilde \alpha (G)$
implies that
$G$
is pancyclic or that
$G=K_{\frac{n}{2},\frac{n}{2}}$
, thus extending the result of McDiarmid and Yolov, and generalizing the classic theorem of Bondy.
Publisher
Cambridge University Press (CUP)
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