Geodesic
Neighbor Product Distinguishing Total Colorings of Planar Graphs with Maximum Degree at least Ten
Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 981-999

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A proper [k]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [k] = 1, 2, . . ., k. Let p(u) denote the product of the color on a vertex u and colors on all the edges incident with u. For each edge uv ∈ E(G), if p(u) ≠ p(v), then we say the coloring c distinguishes adjacent vertices by product and call it a neighbor product distinguishing k-total coloring of G. By X″∏(G), we denote the smallest value of k in such a coloring of G. It has been conjectured by Li et al. that Δ(G) + 3 colors enable the existence of a neighbor product distinguishing total coloring. In this paper, by applying the Combinatorial Nullstellensatz, we obtain that the conjecture holds for planar graph with Δ(G) ≥ 10. Moreover, for planar graph G with Δ(G) ≥ 11, it is neighbor product distinguishing (Δ(G) + 2)-total colorable, and the upper bound Δ(G) + 2 is tight.
Keywords: total coloring, neighbor product distinguishing coloring, planar graph 
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     author = {Dong, Aijun and Li, Tong},
     title = {Neighbor {Product} {Distinguishing} {Total} {Colorings} of {Planar} {Graphs} with {Maximum} {Degree} at least {Ten}},
     journal = {Discussiones Mathematicae. Graph Theory},
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Dong, Aijun; Li, Tong. Neighbor Product Distinguishing Total Colorings of Planar Graphs with Maximum Degree at least Ten. Discussiones Mathematicae. Graph Theory, Tome 41 (2021) no. 4, pp. 981-999. http://geodesic.mathdoc.fr/item/DMGT_2021_41_4_a7/