Abstract
In this chapterwe shall continue the study of critical graphs.Critical graphswere first introduced and investigated by G. A. Dirac in his doctoral thesis; he established the basic properties of critical graphs and published the results in the 1950s in a series of papers. The study of critical graphs was continued in the 1960s by G. Hajós, O. Ore, T. Gallai and others. It was believed (G. A. Dirac, private communication to the third author) that the investigation of structural properties of critical graphsmight be of help in proving the four color theorem. However this was too optimistic. Unless P = NP, there is no good characterization of critical graphs with given chromatic number 𝜒 ≥ 4. .
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Stiebitz, M., Schweser, T., Toft, B. (2024). Properties of Critical Graphs. In: Brooks' Theorem. Springer Monographs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-031-50065-7_4
Download citation
DOI: https://doi.org/10.1007/978-3-031-50065-7_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-50064-0
Online ISBN: 978-3-031-50065-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)