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Relations among the fractional chromatic, choice, Hall, and Hall-condition numbers of simple graphs [electronic resource]

Daneshgar, A. (Amir) ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1016/S0012-365X(01)00117-0
  3. Abstract:
  4. Hall's condition for the existence of a proper vertex list-multicoloring of a simple graph G has recently been used to define the fractional Hall and Hall-condition numbers of G, and . Little is known about , but it is known that , where ‘⩽’ means ‘is a subgraph of’ and α(H) denotes the vertex independence number of H. Let and denote the fractional chromatic and choice (list-chromatic) numbers of G. (Actually, Slivnik has shown that these are equal, but we will continue to distinguish notationally between them.) We give various relations among , , , and , mostly notably that , when G is a line graph. We give examples to show that this equality does not necessarily hold when G is not a line graph. Relations among and behavior of the ‘k-fold’ parameters that appear in the definitions of the fractional parameters are also investigated. The k-fold Hall numbers of the claw are determined and from this certain conclusions follow—for instance, that the sequence (h(k)(G)) of k-fold Hall numbers of a graph G is not necessarily subadditive
  5. Keywords:
  7. Source: Discrete Mathematics ; 2001, Volume 241, Issues 1–3,Pages 189–199
  8. URL: http://www.sciencedirect.com/science/article/pii/S0012365X01001170