Abstract
In 1970, Dirac conjectured that for each integer k ≥ 4, there exists a vertex k-critical graph without any critical edge. In this paper, we introduce strict-vertex critically (k − 1)-de-chromatic pair, strict 2-vertex decomposition graph and so on. By studying their relevant properties, we obtain that any vertex k-critical graph without any critical edge generates an infinite family of vertex k-critical graphs without critical edges.
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References
Brown, J.I.: A Vertex Critical Graph without Critical Edges. Discrete Math. 102, 99–101 (1992)
Erdős, P.: On Some Aspects of My Work with Gabriel Dirac. In: Andersen, L.D., Jakobsen, I.T., Thomassen, C., Toft, B., Vestergaard, P.D. (eds.) Graph Thoery in Memory of G.A. Dirac, Annals of Discrete Mathmatics, vol. 41, pp. 111–116. North-Holland, Amsterdam (1989)
Jensen, T.R.: Dense Critical and Vertex-Critical Graphs. Discrete Math. 258, 63–84 (2002)
Jensen, T.R., Royle, G.F.: Small Graphs of Chromatic Number 5: A Computer Search. J. Graph Theory 19, 107–116 (1995)
Jensen, T.R., Toft, B.: Graph Coloring Problems. Wiley, New York (1995)
Lattanzio, J.J.: A Note on a Conjecture of Dirac. Discrete Math. 258, 323–330 (2002)
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Wang, J. (2009). Infinite Family from Each Vertex k-Critical Graph without Any Critical Edge. In: Du, DZ., Hu, X., Pardalos, P.M. (eds) Combinatorial Optimization and Applications. COCOA 2009. Lecture Notes in Computer Science, vol 5573. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02026-1_22
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DOI: https://doi.org/10.1007/978-3-642-02026-1_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-02025-4
Online ISBN: 978-3-642-02026-1
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