Abstract
AbstractA set of vertices in a graph is a Hamiltonian subset if it induces a subgraph containing a Hamiltonian cycle. Kim, Liu, Sharifzadeh, and Staden proved that for large
$d$
, among all graphs with minimum degree
$d$
,
$K_{d+1}$
minimises the number of Hamiltonian subsets. We prove a near optimal lower bound that takes also the order and the structure of a graph into account. For many natural graph classes, it provides a much better bound than the extremal one (
$\approx 2^{d+1}$
). Among others, our bound implies that an
$n$
-vertex
$C_4$
-free graph with minimum degree
$d$
contains at least
$n2^{d^{2-o(1)}}$
Hamiltonian subsets.
Publisher
Cambridge University Press (CUP)
Subject
Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science