Abstract
Some problems and results on the distribution of subgraphs in colour-critical graphs are discussed.
In section 3 arbitrarily largek-critical graphs withn vertices are constructed such that, in order to reduce the chromatic number tok−2, at leastc k n 2 edges must be removed.
In section 4 it is proved that a 4-critical graph withn vertices contains at mostn triangles. Further it is proved that ak-critical graph which is not a complete graph contains a (k−1)-critical graph which is not a complete graph.
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References
G. A. Dirac, A property of 4-chromatic graphs,J. London Math. Soc. 27 (1952), 85–92.
G. A. Dirac, Some theorems on abstract graphs,Proc. London Math. Soc. 2 (1952), 69–81.
T. Gallai, Kritische Graphen I,Magyar Tud. Akad. Mat. Kutató Int. Közl. 8 (1963), 165–192.
D. Greenwell andL. Lovász, Applications of product colouring,Acta Math. Acad. Sci. Hungar. 25 (1974), 335–340.
G. Hajós, Űber eine Konstruktion nichtn-färbbarer Graphen,Wiss. Z. Martin Luther Univ. Halle-Wittenberg Math. Nat. X/1 (1961), 116–117.
J. B. Kelly andL. M. Kelly, Paths and circuits in critical graphs,Amer. J. Math. 76 (1954), 786–792.
L. Lovász,Combinatorial Problems and Exercises, Akad. Kiadó, Budapest 1979.
F. Nielsen andB. Toft, On a class of 4-chromatic graphs due to T. Gallai,in: Recent advances in Graph Theory, Proceedings of the Symposium held in Prague, June 1974, Acad. Prague 1975, 425–430.
H. Sachs andM. Stiebitz, Construction of colour-critical graphs with given major-vertex subgraphs,Annals of Discr. Math. 17 (1983), 581–598.
M. Simonovits, A method for solving extremal problems in graph theory, stability problems, in:Theory of Graphs, Proc. Colloq. Tihany, 1966, Academic Press, New York, 1968.
B. Toft, On the maximal number of edges of criticalk-chromatic graphs,Studia Sci. Math. Hung. 5 (1970), 461–470.
B. Toft, On critical subgraphs of colour-critical graphs,Discrete Math. 7 (1974), 377–392.
B. Toft, Color-critical graphs and hypergraphs,J. Comb. Theory Ser. B 16 (1974), 145–161.
B. Toft, Graph colouring: A survey of some problems and results,in: Proc. of the Second Danish-Polish Math. Programming Seminar DAPS 79, DIKU Univ. of Copenhagen, Report. No80, April 1980, 351–363.
Z. Tuza andV. Rödl, On colour critical graphs,J. Comb. Th. B,38 (1985), 204–213.
H. J. Voss, Graphs with prescribed maximal subgraphs and critical chromatic graphs,Comment. Math. Univ. Carol. 18 (1977), 129–142.