scholarly journals Neighbor sum distinguishing total chromatic number of planar graphs without 5-cycles

2020 ◽  
Vol 40 (1) ◽  
pp. 243
Author(s):  
Chang-Qing Xu ◽  
Xue Zhao
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Total Chromatic Number

On total chromatic number of planar graphs without 4-cycles

2007 ◽  
Vol 50 (1) ◽  
pp. 81-86 ◽  
Author(s):  
Ying-qian Wang ◽  
Min-le Shangguan ◽  
Qiao Li
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Total Chromatic Number

scholarly journals List edge and list total colorings of planar graphs without non-induced 7-cycles

2013 ◽  
Vol Vol. 15 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Aijun Dong ◽  
Guizhen Liu ◽  
Guojun Li
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Total Chromatic Number ◽  
International Audience

Graphs and Algorithms International audience Giving a planar graph G, let χ'l(G) and χ''l(G) denote the list edge chromatic number and list total chromatic number of G respectively. It is proved that if G is a planar graph without non-induced 7-cycles, then χ'l(G)≤Δ(G)+1 and χ''l(G)≤Δ(G)+2 where Δ(G)≥7.

scholarly journals Total Colorings of $F_5$-Free Planar Graphs with Maximum Degree 8

10.37236/3303 ◽  
2014 ◽  
Vol 21 (1) ◽  
Author(s):  
Jian Chang ◽  
Jian-Liang Wu ◽  
Hui-Juan Wang ◽  
Zhan-Hai Guo
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Total Chromatic Number ◽  
Minimum Number ◽  
Fan Graph

The total chromatic number of a graph $G$, denoted by $\chi′′(G)$, is the minimum number of colors needed to color the vertices and edges of $G$ such that no two adjacent or incident elements get the same color. It is known that if a planar graph $G$ has maximum degree $\Delta ≥ 9$, then $\chi′′(G) = \Delta + 1$. The join $K_1 \vee P_n$ of $K_1$ and $P_n$ is called a fan graph $F_n$. In this paper, we prove that if $G$ is a $F_5$-free planar graph with maximum degree 8, then $\chi′′(G) = 9$.

Neighbor sum distinguishing total chromatic number of planar graphs with maximum degree 10

2017 ◽  
Vol 314 ◽  
pp. 456-468 ◽  
Author(s):  
Donglei Yang ◽  
Lin Sun ◽  
Xiaowei Yu ◽  
Jianliang Wu ◽  
Shan Zhou
Keyword(s):  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Total Chromatic Number

The structure of plane graphs with independent crossings and its applications to coloring problems

2013 ◽  
Vol 11 (2) ◽  
Author(s):  
Xin Zhang ◽  
Guizhen Liu
Keyword(s):  
Planar Graph ◽  
Chromatic Number ◽  
Planar Graphs ◽  
Maximum Degree ◽  
Minimum Degree ◽  
Plane Graph ◽  
Plane Graphs ◽  
Total Chromatic Number ◽  
Linear Arboricity

AbstractIf a graph G has a drawing in the plane in such a way that every two crossings are independent, then we call G a plane graph with independent crossings or IC-planar graph for short. In this paper, the structure of IC-planar graphs with minimum degree at least two or three is studied. By applying their structural results, we prove that the edge chromatic number of G is Δ if Δ ≥ 8, the list edge (resp. list total) chromatic number of G is Δ (resp. Δ + 1) if Δ ≥ 14 and the linear arboricity of G is ℈Δ/2⌊ if Δ ≥ 17, where G is an IC-planar graph and Δ is the maximum degree of G.

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