On fractional ( f , n ) -critical graphs

• Published in: Information processing letters Vol. 109; no. 14; pp. 811 - 815
• Main Authors: Zhou, Sizhong, Shen, Qiqing
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 30.06.2009; Elsevier; Elsevier Sequoia S.A
• Subjects: Algorithmics. Computability. Computer arithmetics; Applied sciences; Binding number; Combinatorial problems; Computer science; control theory; systems; Exact sciences and technology; Fractional [formula omitted]-critical graph; Fractional f-factor; Graph; Graph algorithms; Information retrieval. Graph; Miscellaneous; Optimization; Studies; Theoretical computing; Fractional f-factor; Combinatorial problems; Binding number; Graph; Fractional ( f , n ) -critical graph; Vertex; Edge(graph); Computer theory; Combinatorial problem; Fractional (f,n)-critical graph; Critical graph; Integer; Information processing; Algorithm analysis
• ISSN: 0020-0190; 1872-6119
• DOI: 10.1016/j.ipl.2009.03.026

Let G be a graph of order p, and let a , b and n be nonnegative integers with b ⩾ a ⩾ 2 , and let f be an integer-valued function defined on V ( G ) such that a ⩽ f ( x ) ⩽ b for all x ∈ V ( G ) . A fractional f-factor is a function h that assigns to each edge of a graph G a number in [ 0 , 1 ] , so that for each vertex x we have d G h ( x ) = f ( x ) , where d G h ( x ) = ∑ e ∋ x h ( e ) (the sum is taken over all edges incident to x) is a fractional degree of x in G. Then a graph G is called a fractional ( f , n ) -critical graph if after deleting any n vertices of G the remaining graph of G has a fractional f-factor. The binding number bind ( G ) is defined as follows, bind ( G ) = min { | N G ( X ) | | X | : ∅ ≠ X ⊆ V ( G ) , N G ( X ) ≠ V ( G ) } . In this paper, it is proved that G is a fractional ( f , n ) -critical graph if bind ( G ) > ( a + b − 1 ) ( p − 1 ) ( a p − ( a + b ) − b n + 2 ) and p ⩾ ( a + b ) ( a + b − 3 ) a + b n ( a − 1 ) . Furthermore, it is showed that the result in this paper is best possible in some sense.