Permanent Index of Matrices Associated with Graphs

A total weighting of a graph G is a mapping f which assigns to each element z ∈ V (G)∪E(G) a real number f(z) as its weight. The vertex sum of v with respect to f is φf (v) = ∑ e∈E(v) f(e) + f(v). A total weighting is proper if φf (u) 6= φf (v) for any edge uv of G. A (k, k′)-list assignment is a mapping L which assigns to each vertex v a set L(v) of k permissible weights, and assigns to each edge e a set L(e) of k′ permissible weights. We say G is (k, k′)-choosable if for any (k, k′)-list assignment L, there is a proper total weighting f of G with f(z) ∈ L(z) for each z ∈ V (G)∪E(G). It was conjectured in [T. Wong and X. Zhu, Total weight choosability of graphs, J. Graph Theory 66 (2011), 198–212] that every graph is (2, 2)-choosable and every graph with no isolated edge is (1, 3)-choosable. A promising tool in the study of these conjectures is Combinatorial Nullstellensatz. This approach leads to conjectures on the permanent indices of matrices AG and BG associated to a graphG. In this paper, we establish a method that reduces the study of permanent of matrices associated to a graph G to the study of permanent of matrices associated to induced subgraphs of G. Using this reduction method, we show that if G is a subcubic graph, or a 2-tree, or a Halin graph, or a grid, then AG has permanent index 1. As a consequence, these graphs are (2, 2)-choosable.