Neighbor Product Distinguishing Total Coloring via Combinatorial Nullstellensatz

Let G = (V, E) be a simple graph and ϕ: V(G) ⋃ E(G) → {1, 2, ⋯, k} be a proper total-k-coloring of G. Let f(v) = ϕ(v)Π_uv∈E(G)ϕ(uv). The coloring ϕ is neighbor product distinguishing if f(u) ≠ f(v) for each edge uv ∈ E(G). The neighbor product distinguishing total chromatic number of G, denoted by χ ^″_Π (G), is the smallest integer k such that G admits a k-neighbor product distinguishing total coloring. Li et al. conjectured that χ ^″_Π (G) ≤ Δ(G) + 3 for any graph with at least two vertices and confirmed the conjecture for K₄-minor free graph. In this paper, we prove that for a graph G with at least two vertices, (1) if mad \((G)<{60 \over 17}\), then χ ^″_Π (G) ≤ max{Δ + 2, 8}; (2) if mad \((G)<{8 \over 3}\), then χ ^″_Π (G) ≤ max{Δ + 2, 6}. Furthermore, by using the Combinatorial Nullstellensatz, we simplify their proof and show that χ ^″_Π (G) ≤ max{Δ(G) + 2, 6} for any K₄-minor free graph.