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The strong chromatic number of a graph. (English) Zbl 0751.05034
Let \(G=(V,E)\) be a graph of \(n\) vertices. If \(k\) divides \(n\), then \(G\) is said to be strongly \(k\)-colorable if for any partition of \(V\) into pairwise disjoint sets \(V_ i\), each of cardinality \(k\), there is a proper \(k\)-vertex coloring of \(G\) in which each color meets \(V_ i\), \(i=1,2,\ldots,n/k\), in exactly one vertex. If \(k\) does not divide \(n\), then \(G\) is said to be strongly \(k\)-colorable if the graph obtained from \(G\) by adding to it \(k\lceil n/k\rceil-n\) isolated vertices is strongly \(k\)-colorable. The author shows that there is an absolute constant \(c\) such that for any graph with vertex maximum degree \(d\), the strongly chromatic number is less than or equal to \(cd\).

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