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A graph G is called (r, s)-edge-choosable if for every assignment of sets of size r to the edges of G it is possible to choose for every edge an s-element subset from its set such that the subsets chosen for any pair of adjacent edges are disjoint. The list fractional chromatic index of a graph G is the infimum of all numbers r/s for which G is (r, s)-edge-choosable. Mohar [http://www.fmf.uni-lj.si/~mohar/(June 2003)] posed a problem asking whether every cubic graph is (7,2)-edge-choosable. In 2009, Cranston and West [SIAM J. Discrete Math., 23 (2009), pp. 872-881] showed that every 3-edge-colorable cubic graph is (7,2)-edge-choosable and gave a sufficient condition with the help of which they proved that many non-3-edge-colorable-cubic graphs are (7,2)-edge-choosable. In this paper we prove that every bridgeless cubic graph is (7,2)-edge-choosable. We show that this result cannot be improved in the family of all cubic graphs, in the sense that there exists a cubic graph with list fractional chromatic index 7/2. The original question of Mohar remains open, and we further pose several related problems. [ABSTRACT FROM AUTHOR]

Let G be an undirected graph with a set of vertices V and a set of edges E. Given an integer r, we assign at most r labels (representing "resources") to each vertex. We say that such an assignment is an r-configuration if, for each label c, the vertices labeled by c form a dominating set for G. In this paper, we are interested in the maximum number, Dr(G), of labels that can be assigned to the vertices of graph G by an r-configuration. The decision problem "D1(G) ≥ K?" (known as the domatic number problem) is NP-complete. We first investigate Dr(G) for general graphs and establish bounds on Dr(G) in terms of D1(G) and the minimum vertex degree of G. We then discuss Dr(G) for d-regular graphs. We clearly have Dr(G) ≤ r(d + 1). We show that the problem of testing if Dr(G) = r(d + 1) is solvable in polynomial time for d-regular graphs with ∣V∣ = 2(d + 1), but is NP-complete for those with ∣V∣ = a(d + 1) for some integer a ≥ 3. Finally, we discuss cubic (i.e., 3-regular) graphs. It is easy to show 2r ≤ Dr(G) ≤ 4r for cubic graphs. We show that the decision problem for D1(G) = K is co-NP-complete for K = 2 and is NP-complete for K = 4. Although there are many cubic graphs G with D1(G) = 2, surprisingly, every cubic graph has a 2-configuration with five labels, i.e., D2(G) ≥ 5 and such a 2-configuration can be constructed in polynomial time. We use this fact to show Dr(G) ≥. ⌊5r/2⌋ in general. [ABSTRACT FROM AUTHOR]