论文信息 - Covering Two-Edge-Coloured Complete Graphs with Two Disjoint Monochromatic Cycles

Covering Two-Edge-Coloured Complete Graphs with Two Disjoint Monochromatic Cycles

In 1998 Łuczak Rodl and Szemeredi [7] proved, by means of the Regularity Lemma, that there exists n0 such that, for any n ≥ n0 and two-edge-colouring of Kn, there exists a pair of vertex-disjoint monochromatic cycles of opposite colours covering the vertices of Kn. In this paper we make use of an alternative method of finding useful structure in a graph, leading to a proof of the same result with a much smaller value of n0. The proof gives a polynomial-time algorithm for finding the two cycles.

Peter Allen

[1] János Komlós,et al. Blow-up Lemma , 1997, Combinatorics, Probability and Computing.

[2] Paul Erdös,et al. Vertex coverings by monochromatic cycles and trees , 1991, J. Comb. Theory, Ser. B.

[3] András Gyárfás,et al. Vertex coverings by monochromatic paths and cycles , 1983, J. Graph Theory.

[4] Vojtech Rödl,et al. Partitioning Two-Coloured Complete Graphs into Two Monochromatic Cycles , 1998, Comb. Probab. Comput..

[5] E. Szemerédi. Regular Partitions of Graphs , 1975 .

[6] E. Szemerédi. On sets of integers containing k elements in arithmetic progression , 1975 .

[7] Endre Szemerédi,et al. An improved bound for the monochromatic cycle partition number , 2006, J. Comb. Theory, Ser. B.

[8] G. Szekeres,et al. A combinatorial problem in geometry , 2009 .