Authors: C. Xu, X. Zhao Title: Neighbor sum distinguishing total chromatic number of planar graphs without 5-cycles Source: Discussiones Mathematicae Graph Theory Received 05.12.2017, Revised 08.02.2018, Accepted 07.03.2018, doi: 10.7151/dmgt.2122 | |
Abstract: For a given graph G=(V(G), E(G)), a proper total coloring φ : V(G)∪ E(G)→ {1,2,...,k} is neighbor sum distinguishing if f(u) ≠ f(v) for each edge uv ∈ E(G), where f(v) = ∑_{uv∈E(G)}φ(uv) + φ(v), v ∈ V(G). The smallest integer k in such a coloring of G is the neighbor sum distinguishing total chromatic number, denoted by χ''_{&}Sigma;(G). Pil'{s}niak and Wo'{z}niak first introduced this coloring and conjectured that χ_{Σ}''(G)≤ Δ(G)+3 for any graph with maximum degree Δ(G). In this paper, by using the discharging method, we prove that for any planar graph G without 5-cycles, χ''_{&}Sigma;(G)≤\max{Δ(G)+2,10}. The bound Δ(G)+2 is sharp. Furthermore, we get the exact value of χ''_{&}Sigma;(G) if Δ(G)≥ 9. | |
Keywords: neighbor sum distinguishing total coloring, discharging method, planar graph | |
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