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Packing problems in edge-colored graphs. (English) Zbl 0806.05054
Let $$F$$ be a graph edge-colored with $$k$$ colors. A partition (family of disjoint subsets, respectively) $$\{V_ 1,\dots,V_ m\}$$ of the vertex set of a complete graph edge-colored with $$k$$ colors is an $$F$$-partition ($$F$$-packing, respectively) if each $$V_ i$$ contains a spanning copy of $$F$$ (with the same color pattern as $$F$$). It is shown in the paper that the problem of existence of an $$F$$-partition is NP-complete unless $$F$$ consists of isolated vertices and edges, or $$k=2$$ and $$F$$ is properly 2- edge-colored. It is noted that the case when $$F$$ is an isolated edge is a familiar matching problem and, hence, in this case, the problem is in P. Necessary and sufficient conditions for the existence of an $$F$$-packing are also given for $$F$$ being a properly colored path of length 2. Moreover, a polynomial time algorithm is described to test them. Similar results are proved for $$F$$ consisting of two isolated and differently colored edges. Packings are considered in a more general setting where, instead of one graph $$F$$, we have a family $$\mathcal F$$ of $$k$$-edge-colored graphs. Specifically, sufficient conditions are provided that guarantee existence of large packings of (some) families of trees and forests.

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.) 05C15 Coloring of graphs and hypergraphs
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