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Many Hamiltonian subsets in large graphs with given density
- Part of
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- Journal:
- Combinatorics, Probability and Computing / Volume 33 / Issue 1 / January 2024
- Published online by Cambridge University Press:
- 02 October 2023, pp. 110-120
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- Article
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A 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.
