Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles
Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 243-253
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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) = Σ_ 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_Σ^” (G). Pilśniak and Woźniak 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_Σ^” (G) ≤max{Δ (G)+2, 10 }. The bound Δ (G) + 2 is sharp. Furthermore, we get the exact value of χ_Σ^” (G) if Δ (G) ≥ 9.
Keywords:
neighbor sum distinguishing total coloring, discharging method, planar graph
@article{DMGT_2020_40_1_a15,
author = {Zhao, Xue and Xu, Chang-Qing},
title = {Neighbor {Sum} {Distinguishing} {Total} {Chromatic} {Number} of {Planar} {Graphs} without {5-Cycles}},
journal = {Discussiones Mathematicae. Graph Theory},
pages = {243--253},
publisher = {mathdoc},
volume = {40},
number = {1},
year = {2020},
language = {en},
url = {http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a15/}
}
TY - JOUR AU - Zhao, Xue AU - Xu, Chang-Qing TI - Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles JO - Discussiones Mathematicae. Graph Theory PY - 2020 SP - 243 EP - 253 VL - 40 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a15/ LA - en ID - DMGT_2020_40_1_a15 ER -
%0 Journal Article %A Zhao, Xue %A Xu, Chang-Qing %T Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles %J Discussiones Mathematicae. Graph Theory %D 2020 %P 243-253 %V 40 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a15/ %G en %F DMGT_2020_40_1_a15
Zhao, Xue; Xu, Chang-Qing. Neighbor Sum Distinguishing Total Chromatic Number of Planar Graphs without 5-Cycles. Discussiones Mathematicae. Graph Theory, Tome 40 (2020) no. 1, pp. 243-253. http://geodesic.mathdoc.fr/item/DMGT_2020_40_1_a15/