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Some conditions for the existence of f-factors

✍ Scribed by P. Katerinis

Publisher
John Wiley and Sons
Year
1985
Tongue
English
Weight
268 KB
Volume
9
Category
Article
ISSN
0364-9024

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✦ Synopsis

Let m, 1, n be three odd integers such that m < I < n. It is proved that if a graph G has an mfactor and an rrfactor, then it also has an /factor. In addition, we obtain sufficient conditions for the existence of an f-factor, in terms of vertexdeleted subgraphs.

All graphs considered here are multigraphs (with loops) apd finite. We refer the reader to [2] for standard graph-theoretic terms not defined in this paper. Let G be a graph. Given a function f: V(G) + N = (0, 1,2, . . .}, we say that G has anf-factor if there exists a spanning subgraph H of G such that d&) = f ( x ) for

Iff is the constant function taking the value k then anf-factor is said to be a k-factor. Thus a k-factor of G is a k-regular spanning subgraph of G .

If S C V(G) and H C G\S then e(H, S

) denotes the number of edges having one end vertex in graph H and the other in set S. Given an ordered pair (D, S)

We say that C is an odd or even component of (G\D)\S according to whether dD,S;C) is odd or even. The number of odd components of (G\D)\S is denoted by qG(D, S;f). Tutte I51 proved the following theorem: Tutte'sf-Factor Theorem. A graph G has anf-factor if and only if qc(D, S;f) + c W) -dGv)(x)) 2s c f(x) X C S X E D for all sets D,S C_ V(G), D n S = 8.

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