Abstract
In this paper we consider the problem of partitioning complete multipartite graphs with edges colored by 2 colors into the minimum number of vertex disjoint monochromatic cycles, paths and trees, respectively. For general graphs we simply address the decision version of these three problems the 2-PGMC, 2-PGMP and 2-PGMT problems, respectively. We show that both 2-PGMC and 2-PGMP problems are NP-complete for complete multipartite graphs and the 2-PGMT problem is NP-complete for bipartite graphs. This also implies that all these three problems are NP-complete for general graphs, which solves a question proposed by the authors in a previous paper. Nevertheless, we show that the 2-PGMT problem can be solved in polynomial time for complete multipartite graphs.
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Jin, Z., Kano, M., Li, X. et al. Partitioning 2-edge-colored complete multipartite graphs into monochromatic cycles, paths and trees. J Comb Optim 11, 445–454 (2006). https://doi.org/10.1007/s10878-006-8460-7
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DOI: https://doi.org/10.1007/s10878-006-8460-7