# On Hamiltonian cycles in balanced $k$-partite graphs

## TL;DR

This paper establishes minimum degree conditions for balanced $k$-partite graphs to contain Hamiltonian cycles, providing tight bounds and characterizations of exceptions for all $k \\geq 2$ and divisible $n \\geq 3$.

## Contribution

It introduces precise degree thresholds ensuring Hamiltonicity in balanced $k$-partite graphs, extending and tightening previous results with exact characterizations of special cases.

## Key findings

- Hamiltonian cycles exist under specified degree conditions
- Characterization of non-Hamiltonian graphs when $k=2$ or $k=n/2$ and 4 divides $n$
- Bounds are proven to be tight for all applicable $k$ and $n$

## Abstract

For all integers $k$ with $k\geq 2$, if $G$ is a balanced $k$-partite graph on $n\geq 3$ vertices with minimum degree at least \[ \left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}=\begin{cases} \lceil\frac{n}{2}\rceil+\lfloor\frac{n+2}{k+1}\rfloor-\frac{n}{k} & : k \text{ odd }\\ \frac{n}{2}+\lfloor\frac{n+2}{k+2}\rfloor-\frac{n}{k} & : k \text{ even } \end{cases}, \] then $G$ has a Hamiltonian cycle unless $k=2$ and 4 divides $n$, or $k=\frac{n}{2}$ and 4 divides $n$. In the case where $k=2$ and 4 divides $n$, or $k=\frac{n}{2}$ and 4 divides $n$, we can characterize the graphs which do not have a Hamiltonian cycle and see that $\left\lceil\frac{n}{2}\right\rceil+\left\lfloor\frac{n+2}{2\lceil\frac{k+1}{2}\rceil}\right\rfloor-\frac{n}{k}+1$ suffices. This result is tight for all $k\geq 2$ and $n\geq 3$ divisible by $k$.