Abstract
A graph is IC-planar if it admits a drawing in the plane such that each edge is crossed at most once and two crossed edges share no common end-vertex. A proper total-k-coloring of G is called neighbor sum distinguishing if ∑c(u) ≠ ∑c(v) for each edge uv ∈ E(G), where ∑c(v) denote the sum of the color of a vertex v and the colors of edges incident with v. The least number k needed for such a total coloring of G, denoted by X′′∑(G), is the neighbor sum distinguishing total chromatic number. Pilśniak and Woźniak conjectured X′′∑(G) ≤ △(G) + 3 for any simple graph with maximum degree △(G). By using the famous Combinatorial Nullstellensatz, we prove that above conjecture holds for any triangle free IC-planar graph with △(G) ≥ 7. Moreover, it holds for any triangle free planar graph with △(G) ≥ 6.
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The authors would like to express their thanks to the referees for their invaluable comments and suggestions.
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Supported by National Natural Science Foundation of China (Grant No. 11771443) and the Foundation of Innovative Science and Technology for Youth in Universities of Shandong Province, China (Grant No. 2019KJI001)
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Song, W.Y., Duan, Y.Y. & Miao, L.Y. Neighbor Sum Distinguishing Total Coloring of Triangle Free IC-planar Graphs. Acta. Math. Sin.-English Ser. 36, 292–304 (2020). https://doi.org/10.1007/s10114-020-9189-4
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DOI: https://doi.org/10.1007/s10114-020-9189-4
Keywords
- Neighbor sum distinguishing total coloring
- Combinatorial Nullstellensatz
- triangle free IC-planar graph
- maximum degree