Some results about f-critical graphs

An f-coloring of a multigraph G is a coloring of the edges of E such that each color appears at each vertex v is an element of V at most f(v) times. The minimum number of colors needed to f-color G is called the f-chromatic index of G and is denoted by chi(f)'(G). Various scheduling problems on networks are reduced to finding an f-coloring of a multigraph. Any simple graph G has f-chromatic index equal to Delta(f)(G) or Delta(f)(G)+1,where Delta(f)(G) = max(v is an element of V){[d(v)/f(v)]} and d(v) is the degree of vertex v. A connected graph G is called f-critical if chi(f)'(G) = Delta(f)(G)+1 and chi(f)'(G-e) < chi(f)'(G) for any edge e E 9 Some results about f-critical graphs are given. (c) 2007 Wiley Periodicals, Inc.