Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10

Donglei Yang, Lin Sun, Xiaowei Yu, Jianliang Wu and Shan Zhou

Applied Mathematics and Computation, 2017, vol. 314, issue C, 456-468

Abstract: Given a simple graph G, a proper total-k-coloring ϕ:V(G)∪E(G)→{1,2,…,k} is called neighbor sum distinguishing if Sϕ(u) ≠ Sϕ(v) for any two adjacent vertices u, v ∈ V(G), where Sϕ(u) is the sum of the color of u and the colors of the edges incident with u. It has been conjectured by Pilśniak and Woźniak that Δ(G)+3 colors enable the existence of a neighbor sum distinguishing total coloring. The conjecture is confirmed for any graph with maximum degree at most 3 and for planar graph with maximum degree at least 11. We prove that the conjecture holds for any planar graph G with Δ(G)=10. Moreover, for any planar graph G with Δ(G) ≥ 11, Δ(G)+2 colors guarantee such a total coloring, and the upper bound Δ(G)+2 is tight.

Keywords: Neighbor sum distinguishing total coloring; Planar graph; Combinatorial Nullstellensatz; Discharging (search for similar items in EconPapers)
Date: 2017
References: View references in EconPapers View complete reference list from CitEc
Citations: View citations in EconPapers (6)

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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:314:y:2017:i:c:p:456-468

DOI: 10.1016/j.amc.2017.06.002

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