scholarly journals The list coloring and list total coloring of planar graphs with maximum degree at least 7

2020 ◽  
Vol 40 (4) ◽  
pp. 1005
Author(s):  
Bin Liu ◽  
Lin Sun ◽  
Bing Wang ◽  
Jian-Liang Wu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
List Coloring ◽  
List Total Coloring

scholarly journals Planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable

2016 ◽  
Vol Vol. 17 no. 3 (Graph Theory) ◽  
Author(s):  
Marthe Bonamy ◽  
Benjamin Lévêque ◽  
Alexandre Pinlou
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Edge Coloring ◽  
Total Coloring ◽  
List Total Coloring ◽  
List Edge Coloring ◽  
International Audience

International audience For planar graphs, we consider the problems of <i>list edge coloring</i> and <i>list total coloring</i>. Edge coloring is the problem of coloring the edges while ensuring that two edges that are adjacent receive different colors. Total coloring is the problem of coloring the edges and the vertices while ensuring that two edges that are adjacent, two vertices that are adjacent, or a vertex and an edge that are incident receive different colors. In their list extensions, instead of having the same set of colors for the whole graph, every vertex or edge is assigned some set of colors and has to be colored from it. A graph is minimally edge or total choosable if it is list $\Delta$-edge-colorable or list $(\Delta +1)$-total-colorable, respectively, where $\Delta$ is the maximum degree in the graph. It is already known that planar graphs with $\Delta \geq 8$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable (Li Xu 2011), and that planar graphs with $\Delta \geq 7$ and no triangle sharing a vertex with a $C_4$ or no triangle adjacent to a $C_k (\forall 3 \leq k \leq 6)$ are minimally total colorable (Wang Wu 2011). We strengthen here these results and prove that planar graphs with $\Delta \geq 7$ and no triangle adjacent to a $C_4$ are minimally edge and total choosable.

List edge and list total coloring of planar graphs with maximum degree 8

2015 ◽  
Vol 32 (1) ◽  
pp. 188-197 ◽  
Author(s):  
Huijuan Wang ◽  
Bin Liu ◽  
Xin Zhang ◽  
Lidong Wu ◽  
Weili Wu ◽  
...  
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
List Total Coloring

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

List edge and list total coloring of planar graphs without intersecting 8-cycles

2020 ◽  
Vol 23 (4) ◽  
pp. 925-934
Author(s):  
Wenwen Zhang ◽  
Ye Li
Keyword(s):  
Planar Graphs ◽  
Total Coloring ◽  
List Total Coloring

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

2012 ◽  
Vol 42 (2) ◽  
pp. 151-164 ◽  
Author(s):  
WeiFan WANG ◽  
DanJun HUANG
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
Adjacent Vertex ◽  
Large Maximum

scholarly journals A Sufficient Condition for Planar Graphs of Maximum Degree 6 to be Totally 7-Colorable

2020 ◽  
Vol 2020 ◽  
pp. 1-8
Author(s):  
Enqiang Zhu ◽  
Yongsheng Rao
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Simple Graph ◽  
Total Coloring ◽  
Sufficient Condition ◽  
Open Case ◽  
Total Coloring Conjecture

A total k-coloring of a graph is an assignment of k colors to its vertices and edges such that no two adjacent or incident elements receive the same color. The total coloring conjecture (TCC) states that every simple graph G has a total ΔG+2-coloring, where ΔG is the maximum degree of G. This conjecture has been confirmed for planar graphs with maximum degree at least 7 or at most 5, i.e., the only open case of TCC is that of maximum degree 6. It is known that every planar graph G of ΔG≥9 or ΔG∈7,8 with some restrictions has a total ΔG+1-coloring. In particular, in (Shen and Wang, 2009), the authors proved that every planar graph with maximum degree 6 and without 4-cycles has a total 7-coloring. In this paper, we improve this result by showing that every diamond-free and house-free planar graph of maximum degree 6 is totally 7-colorable if every 6-vertex is not incident with two adjacent four cycles or three cycles of size p,q,ℓ for some p,q,ℓ∈3,4,4,3,3,4.

scholarly journals Total coloring of planar graphs with maximum degree 8

2014 ◽  
Vol 522 ◽  
pp. 54-61 ◽  
Author(s):  
Huijuan Wang ◽  
Lidong Wu ◽  
Jianliang Wu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring

Total coloring of planar graphs of maximum degree eight

2010 ◽  
Vol 110 (8-9) ◽  
pp. 321-324 ◽  
Author(s):  
Nicolas Roussel ◽  
Xuding Zhu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring
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