Abstract: A balanced bipartition of a graph $G(V,E)$ is a bipartition $V_{1}$ and $V_{2}$ of V(G) such that $||V_{1}|-|V_{2}||\leq 1$. The minimum balanced bipartition problem asks for a balanced bipartition minimizing $e(V_{1},V_{2})$, where $e(V_{1},V_{2})$ is the number of edges joining $V_{1}$ and $V_{2}$. In this paper, for Hamilton plane graphs, we study the relation between upper bounds on the minimum of $e(V_{1},V_{2})$ and $d$, which is the difference between the maximum edge function value and the minimum edge function value of Perfect-inner triangle.

Key words: plane graph, Hamilton cycle, balanced bipartite bipartition