scholarly journals Neighbor product distinguishing total colorings of planar graphs with maximum degree at least ten

2019 ◽  
Author(s):  
Aijun Dong ◽  
Tong Li
Keyword(s):  
Planar Graphs ◽  
Maximum Degree

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 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.

scholarly journals Consecutive Colouring of Oriented Graphs

2021 ◽  
Vol 76 (4) ◽  
Author(s):  
Marta Borowiecka-Olszewska ◽  
Ewa Drgas-Burchardt ◽  
Nahid Yelene Javier-Nol ◽  
Rita Zuazua
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Bipartite Graphs ◽  
Complete Graphs ◽  
Oriented Graphs ◽  
Complete Problem ◽  
Np Complete

AbstractWe consider arc colourings of oriented graphs such that for each vertex the colours of all out-arcs incident with the vertex and the colours of all in-arcs incident with the vertex form intervals. We prove that the existence of such a colouring is an NP-complete problem. We give the solution of the problem for r-regular oriented graphs, transitive tournaments, oriented graphs with small maximum degree, oriented graphs with small order and some other classes of oriented graphs. We state the conjecture that for each graph there exists a consecutive colourable orientation and confirm the conjecture for complete graphs, 2-degenerate graphs, planar graphs with girth at least 8, and bipartite graphs with arboricity at most two that include all planar bipartite graphs. Additionally, we prove that the conjecture is true for all perfect consecutively colourable graphs and for all forbidden graphs for the class of perfect consecutively colourable graphs.

scholarly journals Planar graphs with maximum degree Δ≥9 are (Δ+1)-edge-choosable—A short proof

2010 ◽  
Vol 310 (21) ◽  
pp. 3049-3051 ◽  
Author(s):  
Nathann Cohen ◽  
Frédéric Havet
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Short Proof

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 strong edge coloring of planar graphs with maximum degree 4

2019 ◽  
Vol 342 (5) ◽  
pp. 1471-1480
Author(s):  
Ming Chen ◽  
Jie Hu ◽  
Xiaowei Yu ◽  
Shan Zhou
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Edge Coloring

The linear arboricity of planar graphs of maximum degree seven is four

2008 ◽  
Vol 58 (3) ◽  
pp. 210-220 ◽  
Author(s):  
Jian-Liang Wu ◽  
Yu-Wen Wu
Keyword(s):  
Planar Graphs ◽  
Maximum Degree ◽  
Linear Arboricity
Sign in / Sign up

Export Citation Format

Share Document