A proper total coloring distinguishing adjacent vertices by sums of planar graphs without intersecting triangles

2015 ◽  
Vol 32 (2) ◽  
pp. 626-638 ◽  
Author(s):  
Jihui Wang ◽  
Qiaoling Ma ◽  
Xue Han ◽  
Xiuyun Wang
Keyword(s):  
Planar Graphs ◽  
Total Coloring ◽  
Proper Total Coloring

scholarly journals Neighbor Sum Distinguishing Total Choosability of IC-Planar Graphs without Theta Graphs Θ2,1,2

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 708
Author(s):  
Donghan Zhang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
Disjoint Paths ◽  
Combinatorial Nullstellensatz ◽  
Proper Total Coloring

A theta graph Θ2,1,2 is a graph obtained by joining two vertices by three internally disjoint paths of lengths 2, 1, and 2. A neighbor sum distinguishing (NSD) total coloring ϕ of G is a proper total coloring of G such that ∑z∈EG(u)∪{u}ϕ(z)≠∑z∈EG(v)∪{v}ϕ(z) for each edge uv∈E(G), where EG(u) denotes the set of edges incident with a vertex u. In 2015, Pilśniak and Woźniak introduced this coloring and conjectured that every graph with maximum degree Δ admits an NSD total (Δ+3)-coloring. In this paper, we show that the listing version of this conjecture holds for any IC-planar graph with maximum degree Δ≥9 but without theta graphs Θ2,1,2 by applying the Combinatorial Nullstellensatz, which improves the result of Song et al.

Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 9

2019 ◽  
Vol 342 (5) ◽  
pp. 1392-1402
Author(s):  
Jie Hu ◽  
Guanghui Wang ◽  
Jianliang Wu ◽  
Donglei Yang ◽  
Xiaowei Yu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
Adjacent Vertex

List edge and list total coloring of 1-planar graphs

2012 ◽  
Vol 7 (5) ◽  
pp. 1005-1018 ◽  
Author(s):  
Xin Zhang ◽  
Jianliang Wu ◽  
Guizhen Liu
Keyword(s):  
Planar Graphs ◽  
Total Coloring ◽  
List Total Coloring

scholarly journals A note on the total coloring of planar graphs without adjacent 4-cycles

2012 ◽  
Vol 312 (11) ◽  
pp. 1923-1926 ◽  
Author(s):  
Hui-Juan Wang ◽  
Jian-Liang Wu
Keyword(s):  
Planar Graphs ◽  
Total Coloring
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