scholarly journals A sufficient condition for a planar graph to be class 1

2007 ◽  
Vol 385 (1-3) ◽  
pp. 71-77 ◽  
Author(s):  
Weifan Wang ◽  
Yongzhu Chen
Keyword(s):  
Planar Graph ◽  
Sufficient Condition ◽  
Class 1

scholarly journals A sufficient condition for a planar graph to be4-choosable

2017 ◽  
Vol 224 ◽  
pp. 120-122 ◽  
Author(s):  
Renyu Xu ◽  
Jian-Liang Wu
Keyword(s):  
Planar Graph ◽  
Sufficient Condition

A sufficient condition for a planar graph to be 3-colorable

2013 ◽  
Vol 43 (4) ◽  
pp. 409-421
Author(s):  
YingQian WANG ◽  
YingLi KANG
Keyword(s):  
Planar Graph ◽  
Sufficient Condition

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 On the edge coloring of graph products

2005 ◽  
Vol 2005 (16) ◽  
pp. 2669-2676 ◽  
Author(s):  
M. M. M. Jaradat
Keyword(s):  
Chromatic Number ◽  
Edge Coloring ◽  
Sufficient Condition ◽  
Graph Products ◽  
Strong Product ◽  
Class 1 ◽  
Minimum Number

The edge chromatic number ofGis the minimum number of colors required to color the edges ofGin such a way that no two adjacent edges have the same color. We will determine a sufficient condition for a various graph products to be of class 1, namely, strong product, semistrong product, and special product.

A sufficient condition for a regular graph to be class 1

1993 ◽  
Vol 17 (6) ◽  
pp. 701-712 ◽  
Author(s):  
A. J. W. Hilton ◽  
Cheng Zhao
Keyword(s):  
Regular Graph ◽  
Sufficient Condition ◽  
Class 1

A sufficient condition for a planar graph to be 3-choosable

2007 ◽  
Vol 104 (4) ◽  
pp. 146-151 ◽  
Author(s):  
Liang Shen ◽  
Yingqian Wang
Keyword(s):  
Planar Graph ◽  
Sufficient Condition
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