Abstract
In our work, we define a k-tuple of positive integers \((x_1, \ldots , x_k)\) to be a \(\chi \)-sequence if there exists a k-chromatic graph G such that for each \(i \in \{1, \ldots , k\}\), the order of a minimum i-chromatic subgraph of G is equal to \(x_i\). Denote by \(\mathcal {X}_k\) the set of all \(\chi \)-sequences of length k. A very difficult question is to determine, for a given \((x_1, \ldots , x_k) \in \mathcal {X}_k\), the set of all integers y such that \((x_1, \ldots , x_k, y) \in \mathcal {X}_{k+1}\). We propose a few variants of this question and elaborate upon a number of partial results along the way.
Similar content being viewed by others
References
Brandt, S., Harant, J., Naumann, S.: On degree sums of a triangle-free graph. Discrete Math. 337, 76–82 (2014)
Chvátal, V.: The minimality of the Mycielski graph, Graphs and Combinatorics (Proc. Capital Conf., George Washington Univ., Washington, D.C.), Lecture Notes in Mathematics 406, Springer-Verlag, pp. 243 – 246 (1973)
Cropper, M., Gyárfás, A., Lehel, J.: Hall ratio of the Mycielski graphs. Discrete Math. 306, 1988–1990 (2006)
Goedgebeur, J.: On minimal triangle-free 6-chromatic graphs. J. Graph Theory 93(1), 34–48 (2020)
Hajós, G.: Über eine konstruktion nicht \(n\)-färbbarer graphen. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg Math.-Natur. Reihe 10, 116–117 (1961)
Jensen, T.R., Toft, B.: Graph coloring problems. Wiley, Hoboken (1995)
Jensen, T., Royle, G.F.: Small graphs with chromatic number 5: a computer search. J. Graph Theory 19(1), 107–116 (1995)
Müller, T., Stehlik, M.: Generalized Mycielski graphs and the Borsuk-Ulam theorem. Electron. J. Comb. 26(4), #P4.8 (2019)
Mycielski, J.: Sur le coloriage des graphes. Colloq. Math. 3, 161–162 (1955)
Van Ngoc, N., Tuza, Z.: 4-chromatic graphs with large odd girth. Discrete Math. 138, 387–392 (1995)
Sachs, H., Stiebitz, M.: On constructive methods in the theory of colour-critical graphs. Discrete Math. 74, 201–226 (1989)
Stiebitz, M.: Contributions to the theory of color-critical graphs. Diss. B, TH Ilmenau (1985). https://www.tu-ilmenau.de/dma/team/michael-stiebitz/
Acknowledgements
The authors thank Jan Goedgebeur and Gordon Royle for their helpful correspondence, as well as the anonymous referee for a number of useful suggestions and for the time they spent with our paper.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Newman, N., Noble, M. Questions on Color-Critical Subgraphs. Graphs and Combinatorics 37, 313–324 (2021). https://doi.org/10.1007/s00373-020-02243-z
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-020-02243-z