SUM LIST EDGE COLORINGS OF GRAPHS

Let G = (V, E) be a simple graph and for every edge e is an element of E let L(e) be a set (list) of available colors. The graph G is called L-edge colorable if there is a proper edge coloring c of G with c(e) is an element of L(e) for all e is an element of E. A function f : E -> N is called an edge choice function of G and G is said to be f-edge choosable if G is L-edge colorable for every list assignment L with vertical bar L(e)vertical bar - f(e) for all e is an element of E. Set size(f) = Sigma(e is an element of E) f(e) and define the sum choice index chi(sc)' (G) as the minimum of size(f) over all edge choice functions f of G. There exists a greedy coloring of the edges of G which leads to the upper bound chi(sc)' (G) <= 1/2 Sigma(v is an element of V) d(v)(2). A graph is called sec-greedy if its sum choice index equals this upper bound. We present some general results on the sum choice index of graphs including a lower bound and we determine this index for several classes of graphs. Moreover, we present classes of sec-greedy graphs as well as all such graphs of order at most 5.