Relations among the fractional chromatic, choice, Hall, and Hall-condition numbers of simple graphs

Abstract

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, h f(G) and s f(G) . Little is known about h f(G) , but it is known that s f(G)= max[|V(H)|/α(H);H⩽G] , where ‘ ⩽’ means ‘is a subgraph of’ and α(H) denotes the vertex independence number of H. Let χ f(G) and c f(G) 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 χ f(G) , c f(G) , h f(G) , and s f(G) , mostly notably that χ f(G)=c f(G)=s f(G) , 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.