Enumeration of Labelled and Unlabelled Hamiltonian Cycles in Complete $k$-partite Graphs

TL;DR

This paper develops recurrence relations to precisely count labelled and unlabelled Hamiltonian cycles in complete $n$-partite graphs with equal part sizes, advancing combinatorial enumeration methods.

Contribution

It introduces new recurrence relations for exact enumeration of Hamiltonian cycles in complete $k$-partite graphs with equal parts, covering both labelled and unlabelled cases.

Findings

Derived recurrence relations for counting Hamiltonian cycles.

Provided exact enumeration formulas for arbitrary $n$ and $d$.

Enhanced understanding of Hamiltonian cycle enumeration in Turán graphs.

Abstract

We enumerate labelled and unlabelled Hamiltonian cycles in complete $n$-partite graphs $K_{d,d,\ldots,d}$ having exactly $d$ vertices in each part (in other words, Tur\'an graphs $T(nd, n))$. We obtain recurrence relations that allow us to find the exact values $b_{n}^{(d)}$ of such cycles for arbitrary $n$ and $d$