On judicious partitions of graphs

Abstract

Let $k, m$ be positive integers, let $G$ be a graph with $m$ edges, and let $h(m)=\sqrt{2m+\frac{1}{4}}-\frac{1}{2}$. Bollobás and Scott asked whether $G$ admits a $k$-partition $V_{1}, V_{2}, \ldots , V_{k}$ such that $\max _{1\le i\le k} \{e(V_{i})\}\le \frac{m}{k^2}+\frac{k-1}{2k^2}h(m)$ and $e(V_1, \ldots , V_k)\ge {k-1\over k} m +{k-1\over 2k}h(m) -\frac{(k-2)^{2}}{8k}$. In this paper, we present a positive answer to this problem on the graphs with large number of edges and small number of vertices with degrees being multiples of $k$. Particularly, if $d$ is not a multiple of $k$ and $G$ is $d$-regular with $m\ge {9\over 128}k^4(k-2)^2$, then $G$ admits a $k$-partition as desired. We also improve an earlier result by showing that $G$ admits a partition $V_{1}, V_{2}, \ldots , V_{k}$ such that $e(V_{1},V_{2},\ldots ,V_{k})\ge \frac{k-1}{k}m+\frac{k-1}{2k}h(m)-\frac{(k-2)^{2}}{2(k-1)}$ and $\max _{1\le i\le k}\{e(V_{i})\}\le \frac{m}{k^{2}}+\frac{k-1}{2k^{2}}h(m)$.