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Graph partitions with minimum degree constraints. (English) Zbl 0955.05102

Summary: Given a graph with \(n\) nodes and minimum degree \(\delta\), we give a polynomial time algorithm that constructs a partition of the nodes of the graph into two sets \(X\) and \(Y\) such that the sum of the minimum degrees in \(X\) and in \(Y\) is at least \(\delta\) and the cardinalities of \(X\) and \(Y\) differ by at most \(\delta\) (\(\delta+1\) if \(n\not\equiv\delta\pmod 2\)). The existence of such a partition was shown by J. Sheehan [Discrete Math. 68, No. 2/3, 299-307 (1988; Zbl 0644.05030)].

MSC:

05C85 Graph algorithms (graph-theoretic aspects)
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C35 Extremal problems in graph theory

Citations:

Zbl 0644.05030
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References:

[1] Bollobás, B., Extremal Graph Theory (1978), Academic Press: Academic Press New York · Zbl 0419.05031
[2] Lovaśz, L., On decomposition of graphs, Stud. Scient. Math. Hung., 1, 237-238 (1966) · Zbl 0151.33401
[3] Maurer, S. B., Vertex colorings without isolates, J. Combin. Theory Ser. B, 27, 294-319 (1979) · Zbl 0351.05108
[4] Sheehan, J., Graph decomposition with constraints on the minimum degree, Discrete Math., 68, 299-307 (1988) · Zbl 0644.05030
[5] Sheehan, J., Balanced graph with minimum degree constraints, Discrete Math., 102, 307-314 (1992) · Zbl 0770.05094
[6] Sheehan, J., Graphical decompositions, Discrete Math., 125, 347-355 (1994) · Zbl 0795.05079
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