The Existence of Hamilton Cycle in n-Balanced k-Partite Graphs

TL;DR

This paper establishes a minimum edge count condition for n-balanced k-partite graphs to contain a Hamilton cycle, extending known results and potentially providing optimal bounds.

Contribution

It proves a new edge threshold condition ensuring Hamiltonicity in n-balanced k-partite graphs, generalizing previous results and suggesting optimality.

Findings

Derived a minimum edge count for Hamilton cycles in n-balanced k-partite graphs.

Extended classical Hamiltonicity conditions to k-partite graphs with specific bounds.

Proved the bounds are potentially the best possible.

Abstract

Let $G_{k,n}$ be the $n$-balanced $k$-partite graph, whose vertex set can be partitioned into $k$ parts, each has $n$ vertices. In this paper, we prove that if $k \geq 2,n \geq 1$, for the edge set $E(G)$ of $G_{k,n}$ $$|E(G)| \geq\left\{\begin{array}{cc} 1 & \text { if } k=2, n=1 n^{2} C_{k}^{2}-(k-1) n+2 & \text { other } \end{array}\right.$$ then $G_{k,n}$ is hamiltonian. And the result may be the best.