Let G = (V,E) be a graph. For each e ∈E(G) and v ∈V(G), let Le and Lv, respectively, be a list of real numbers. Let w be a function on V(G) ∪E(G) such that w(e) ∈Le for each e ∈E(G) and w(v) ∈Lv for each v ∈V(G), and let cw be the vertex colouring obtained by cw(v) = w(v) + ∑ₑ ∋vw(e). A graph is (k,l)-weight choosable if there exists a weighting function w for which cw is proper whenever |Lv| ≥k and |Le| ≥l for every v ∈V(G) and e ∈E(G). A sufficient condition for a graph to be (1,l)-weight choosable was developed by Bartnicki, Grytczuk and Niwczyk (2009), based on the Combinatorial Nullstellensatz, a parameter which they call the monomial index of a graph, and matrix permanents. This paper extends their method to establish the first general upper bound on the monomial index of a graph, and thus to obtain an upper bound on l for which every admissible graph is (1,l)-weight choosable. Let ∂2(G) denote the smallest value s such that every induced subgraph of G has vertices at distance 2 whose degrees sum to at most s. We show that every admissible graph has monomial index at most ∂2(G) and hence that such graphs are (1, ∂2(G)+1)-weight choosable. While this does not improve the best known result on (1,l)-weight choosability, we show that the results can be extended to obtain improved bounds for some graph products; for instance, it is shown that G □ Kn is (1, nd+3)-weight choosable if G is d-degenerate.
Discrete Mathematics & Theoretical Computer Science