Total coloring of planar graphs of maximum degree eight

The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ ″ ( G ) . It is known that if a planar graph G has maximum degree Δ ⩾ 9 , then χ ″ ( G ) = Δ + 1 . Recently Hou et al. (Graphs and Combinatorics 24 (20...

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Bibliographic Details
Published in:Information processing letters Vol. 110; no. 8; pp. 321 - 324
Main Authors: Roussel, Nicolas, Zhu, Xuding
Format: Journal Article
Language:English
Published: Amsterdam Elsevier B.V 01.04.2010
Elsevier
Elsevier Sequoia S.A
Subjects:
Algorithmics. Computability. Computer arithmetics
Applied sciences
Coloring
Combinatorial analysis
Combinatorial problems
Combinatorics
Combinatorics. Ordered structures
Computer science; control theory; systems
Exact sciences and technology
Graph coloring
Graph theory
Graphs
Information retrieval. Graph
Integers
Mathematics
Miscellaneous
Optimization algorithms
Planar graphs
Sciences and techniques of general use
Studies
Theoretical computing
Total coloring
Total coloring
Combinatorial problems
Planar graphs
Vertex
Edge(graph)
Computer theory
Combinatorial problem
Planar graph
Color
Chromatic number
Integer
Maximum
Information processing
Graph degree
Graph colouring
Combinatorics
Algorithm analysis
ISSN:0020-0190, 1872-6119
Online Access:Get full text
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Summary:The minimum number of colors needed to properly color the vertices and edges of a graph G is called the total chromatic number of G and denoted by χ ″ ( G ) . It is known that if a planar graph G has maximum degree Δ ⩾ 9 , then χ ″ ( G ) = Δ + 1 . Recently Hou et al. (Graphs and Combinatorics 24 (2008) 91–100) proved that if G is a planar graph with maximum degree 8 and with either no 5-cycles or no 6-cycles, then χ ″ ( G ) = 9 . In this Note, we strengthen this result and prove that if G is a planar graph with maximum degree 8, and for each vertex x, there is an integer k x ∈ { 3 , 4 , 5 , 6 , 7 , 8 } such that there is no k x -cycle which contains x, then χ ″ ( G ) = 9 .
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ISSN:0020-0190
1872-6119
DOI:10.1016/j.ipl.2010.02.012