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Coloring of signed graphs / by Yingli Kang ; Adviser: Prof. Dr. Eckhard Steffen. Paderborn, 2018
Inhalt
Acknowledgements
Contents
List of Figures
1 Introduction
1.1 Signed graphs, balances and switchings
1.2 Colorings of signed graphs
2 Circular chromatic number c of signed graphs
2.1 (k,d)-colorings of a signed graph
2.1.1 A recoloring technique: Updating
2.1.2 Basic properties of (k,d)-colorings
2.1.3 c((G,)): from infimum to minimum
2.2 Circular r-colorings of a signed graph
2.3 Relation between c((G,)) and ((G,))
2.3.1 Relations in general
2.3.2 Signed graphs for which c=-1
2.3.3 Improved lower bound
2.4 Signed graphs for which c=
2.4.1 Interval colorings of signed graphs
2.4.2 Construction
3 Chromatic number of signed graphs
3.1 Some basic properties
3.2 Chromatic spectrum of a graph
3.2.1 Determination of m(G)
3.2.2 Chromatic critical signed graphs
3.2.3 Proof of Theorem 3.6
3.3 An analogue of Brooks' Theorem for signed graphs
3.4 First Hajós-like theorem for signed graphs
3.4.1 Introduction
3.4.2 Signed bi-graphs
3.4.3 Graph operations on signed bi-graphs
3.4.4 Useful lemmas
3.4.5 Proof of the theorem
4 Signed chromatic number of signed graphs
4.1 Preliminary
4.2 Signed chromatic spectrum of a graph
4.2.1 Signed chromatic critical graph
4.2.2 Proof of Theorem 4.4
4.3 Second Hajós-like theorem for signed graphs
4.4 Relation between and
5 Choosability in signed graphs
5.1 Definitions and basic properties
5.2 Choosability in signed planar graphs
5.2.1 5-choosability
5.2.2 4-choosability
5.2.3 3-choosability
6 3-colorings of planar graphs
6.1 Introduction
6.2 Notations and formulation of the main theorem
6.3 Proof of the main theorem
6.3.1 Structural properties of the minimal counterexample G
6.3.2 Discharging in G
7 Conclusion and future work
Bibliography
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