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Transformation of a graph into a bipartite graph by splitting vertices

✍ Scribed by O. F. Cherepov


Publisher
Springer US
Year
1976
Tongue
English
Weight
185 KB
Volume
11
Category
Article
ISSN
1573-8337

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For two integers a and b, we say that a bipartite graph G admits an (a, b)bipartition if G has a bipartition (X, Y ) such that |X| = a and |Y | = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit (a, b)-bipartitions. In this paper, we prove

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Wang, H., Partition of bipartite graph into cycles, Discrete Mathematics 117 (1993) 287-291. El-Zahar (1984) conjectured that if G is a graph on n, + n, + + nk vertices with ni > 3 for 1s i < k and minimum degree 6(G)>rn,/21+rn2/21+ ... +rn,/21, then G contains k vertex-disjoint cycles of lengths n,

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For two integers a and b, we say that a bipartite graph G admits an ( a , b)-bipartition if G has a bipartition ( X , Y ) such that /XI = a and ( Y / = b. We say that two bipartite graphs G and H are compatible if, for some integers a and b, both G and H admit ( a , b)-bipartitions. In this note, w

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We prove the following theorem. "I'neorem. If G is a balanced bipartite graph with bipartition (A, B), [A I = IBI = n, such that for any x ~ A, y ~ B, d(x) + d(y) >>-n + 2, then for any (nl, n2), ni >I 2, n -----n I + hE, G contains two independent cycles of lengths 2nl and 2n2.