Bounds for Judicious Balanced Bipartitions of Graphs

Abstract

A bipartition of the vertex set of a graph is called balanced if the sizes of the sets in the bipartition differ by at most one. Bollob$\acute{a}$s and Scott proved that every regular graph with m edges admits a balanced bipartition $V_{1}$, $V_{2}$ of V(G) such that $\max \{e(V_{1}), e(V_{2}) \}< \frac{m}{4}$. Only allowing $\varDelta (G)-\delta (G)$ =1 and 2, Yan and Xu, and Hu, He and Hao, respectively showed that a graph G with n vertices and m edges has a balanced bipartition $V_{1}$, $V_{2}$ of V(G) such that $\max \{e(V_{1}), e(V_{2}) \}\le \frac{m}{4}+O(n)$. In this paper, we give an upper bound for balanced bipartition of graphs G with $\varDelta (G)-\delta (G)=t-1$, $t\ge 2$ is an integer. Our result extends the conclusions above.