The adjacent vertex distinguishing total choosability of planar graphs with maximum degree at least eleven

2017 ◽  
Vol 35 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Xiaohan Cheng ◽  
Jianliang Wu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Adjacent Vertex

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

Adjacent vertex distinguishing total coloring of planar graphs with maximum degree 8

2020 ◽  
Vol 343 (10) ◽  
pp. 112014
Author(s):  
Yulin Chang ◽  
Jie Hu ◽  
Guanghui Wang ◽  
Xiaowei Yu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Total Coloring ◽  
Adjacent Vertex

scholarly journals Clustered 3-colouring graphs of bounded degree

2021 ◽  
pp. 1-13
Author(s):  
Vida Dujmović ◽  
Louis Esperet ◽  
Pat Morin ◽  
Bartosz Walczak ◽  
David R. Wood
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bounded Degree ◽  
Bounded Number ◽  
Proper Vertex ◽  
Vertex Colouring

Abstract A (not necessarily proper) vertex colouring of a graph has clustering c if every monochromatic component has at most c vertices. We prove that planar graphs with maximum degree $\Delta$ are 3-colourable with clustering $O(\Delta^2)$ . The previous best bound was $O(\Delta^{37})$ . This result for planar graphs generalises to graphs that can be drawn on a surface of bounded Euler genus with a bounded number of crossings per edge. We then prove that graphs with maximum degree $\Delta$ that exclude a fixed minor are 3-colourable with clustering $O(\Delta^5)$ . The best previous bound for this result was exponential in $\Delta$ .

scholarly journals Adjacent vertex distinguishing indices of planar graphs without 3-cycles

2015 ◽  
Vol 338 (3) ◽  
pp. 139-148 ◽  
Author(s):  
Danjun Huang ◽  
Zhengke Miao ◽  
Weifan Wang
Keyword(s):  
Planar Graphs ◽  
Adjacent Vertex

scholarly journals Planar Graphs of Maximum Degree Six without 7-cycles are Class One

10.37236/2589 ◽  
2012 ◽  
Vol 19 (3) ◽  
Author(s):  
Danjun Huang ◽  
Weifan Wang
Keyword(s):  
Planar Graph ◽  
Planar Graphs ◽  
Maximum Degree

In this paper, we prove that every planar graph of maximum degree six without 7-cycles is class one.

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