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A NOTE ON ALMOST BALANCED BIPARTITIONS OF A GRAPH

Part of: Graph theory

Published online by Cambridge University Press:  14 October 2014

XIAOLAN HU ,
YUNQING ZHANG  and
YAOJUN CHEN
XIAOLAN HU
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, China
YUNQING ZHANG
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, China
YAOJUN CHEN*
Affiliation:
Department of Mathematics, Nanjing University, Nanjing 210093, China email [email protected]
*
✉[email protected]
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Abstract

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Let $G$ be a graph of order $n\geq 6$ with minimum degree ${\it\delta}(G)\geq 4$. Arkin and Hassin [‘Graph partitions with minimum degree constraints’, Discrete Math. 190 (1998), 55–65] conjectured that there exists a bipartition $S,T$ of $V(G)$ such that $\lfloor n/2\rfloor -2\leq |S|,|T|\leq \lceil n/2\rceil +2$ and the minimum degrees in the subgraphs induced by $S$ and $T$ are at least two. In this paper, we first show that $G$ has a bipartition such that the minimum degree in each part is at least two, and then prove that the conjecture is true if the complement of $G$ contains no complete bipartite graph $K_{3,r}$, where $r=\lfloor n/2\rfloor -3$.

Keywords

almost balanced bipartitioncomplement

MSC classification

Primary: 05C70: Factorization, matching, partitioning, covering and packing
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 91 , Issue 2 , April 2015 , pp. 177 - 182
Copyright
Copyright © 2014 Australian Mathematical Publishing Association Inc. 

References

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