The Strong Fractional Choice Number and the Strong Fractional Paint Number of Graphs

This paper studies the strong fractional choice number chˢf(G) and the strong fractional paint number ptˢf(G) of a graph G. We prove that these parameters of any finite graph are rational numbers. On the other hand, for any positive integers $p,q$ satisfying 2 ≤ 2p/2q+1 ≤ /q, we construct a graph G with chˢf(G) = ptˢf(G) = p/q. The relationship between ptˢf(G) and chˢf(G) is explored. We prove that the gap ptˢf(G)-chˢf(G) can be arbitrarily large. The strong fractional choice number of a family G of graphs is the supremum of the strong fractional choice numbers of graphs in G. Let P denote the class of planar graphs and Pk₁,…, kq denote the class of planar graphs without kᵢ-cycles for i=1,…, q. We prove that 3 + 1/2 ≤ chˢf(P₄) ≤ 4, chˢf(Pₖ)=4 for k ∈ \5,6\, 3 +1/12 ≤ chˢf(P4,5) ≤ 4, and chˢf(P) ≥ 4+ 13. The last result improves the lower bound 4+ 29 in [Zhu, J. Combin. Theory Ser. B, 122 (2017), pp. 794--799].