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International Journal of Mathematics and Mathematical Sciences
Volume 2011 (2011), Article ID 947151, 7 pages
Research Article

An Intermediate Value Theorem for the Arboricities

1Department of Mathematics, School of Science, University of the Thai Chamber of Commerce, Bangkok 10400, Thailand
2Department of Mathematics, Srinakharinwirot University, Sukhumvit 23, Bangkok 10110, Thailand

Received 26 October 2010; Accepted 29 April 2011

Academic Editor: Aloys Krieg

Copyright © 2011 Avapa Chantasartrassmee and Narong Punnim. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Let be a graph. The vertex (edge) arboricity of denoted by is the minimum number of subsets into which the vertex (edge) set of can be partitioned so that each subset induces an acyclic subgraph. Let be a graphical sequence and let be the class of realizations of . We prove that if , then there exist integers and such that has a realization with if and only if is an integer satisfying . Thus, for an arbitrary graphical sequence and , the two invariants and   naturally arise and hence We write for the degree sequence of an -regular graph of order . We prove that . We consider the corresponding extremal problem on vertex arboricity and obtain in all situations and for all .