An (s, t)-partition of a graph G=(V,E)\documentclass[12pt]{minimal}
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\begin{document}$$G=(V,E)$$\end{document} is a partition of V=V1∪V2\documentclass[12pt]{minimal}
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\begin{document}$$V=V_1\cup V_2$$\end{document} such that δ(G[V1])≥s\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G[V_1])\ge s$$\end{document} and δ(G[V2])≥t\documentclass[12pt]{minimal}
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\begin{document}$$\delta (G[V_2])\ge t$$\end{document}. It has been conjectured that, for sufficiently large n, every d-regular graph of order n has a (⌈d2⌉,⌈d2⌉)\documentclass[12pt]{minimal}
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\begin{document}$$(\lceil \frac{d}{2}\rceil , \lceil \frac{d}{2}\rceil )$$\end{document}-partition (called an internal partition). In this paper, we prove that every d-regular graph of order n has a (⌈d2⌉,⌊d2⌋)\documentclass[12pt]{minimal}
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\begin{document}$$(\lceil \frac{d}{2}\rceil , \lfloor \frac{d}{2}\rfloor )$$\end{document} partition (called a weak internal partition) for d≤9\documentclass[12pt]{minimal}
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\begin{document}$$d\le 9$$\end{document} and sufficiently large n.
