Abstract
The heterochromatic tree partition number of an \(r\)-edge-colored graph \(G,\) denoted by \(t_r(G),\) is the minimum positive integer \(p\) such that whenever the edges of the graph \(G\) are colored with \(r\) colors, the vertices of \(G\) can be covered by at most \(p\) vertex disjoint heterochromatic trees. In this article we determine the upper and lower bounds for the heterochromatic tree partition number \(t_r(K_{n_1,n_2,\ldots ,n_k})\) of an \(r\)-edge-colored complete \(k\)-partite graph \(K_{n_1,n_2,\ldots ,n_k}\), and the gap between upper and lower bounds is at most one.
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Acknowledgments
The authors would like to thank the referees for helpful suggestions. This study was supported by NSFC (10701065 and 11101378), Zhejiang Innovation Project (Grant No. T200905) and ZJNSF (Z6090150).
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Jin, Z., Zhu, P. Heterochromatic tree partition number in complete multipartite graphs. J Comb Optim 28, 321–340 (2014). https://doi.org/10.1007/s10878-012-9557-9
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DOI: https://doi.org/10.1007/s10878-012-9557-9