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Anticritical graphs

  • Frank Harary (a1) and Carsten Thomassen (a2)

Abstract

The term ‘critical graph’ has been used most frequently to mean a graph such that the removal of any line decreases the chromatic number. This can be generalized to define a graph G to be ‘critical (μ)’ for an arbitrary invariant, μ if, for each line e, μ(Ge) ≠ μ(G). Analogously we may consider any line e which joins two points not already adjacent in G and define G as ‘anticritical (μ)’ if μ(G + e) ≠ μ(G). Anticritical graphs are investigated here with respect to the following invariants: chromatic number, line-chromatic number, point-connectivity, line-connectivity, point-independence number, line-independence number, diameter, radius, and cyclability number.

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(1)Beineke, L. W. and Wilson, R. J.On the edge-chromatic number of a graph. Discrete Math. 5 (1973), 1520.
(2)Chvétal, V. New directions in hamiltonian graph theory. New Directions in the Theory of Graphs, ed. Harary, F. (New York, Academic Press, 1973), 6595.
(3)Dirac, G. A.A property of 4-chromatic graphs and some remarks on critical graphs. J. London Math. Soc. 2 (1952), 6981.
(4)Dirac, G. A.In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen. Math. Nachr. 22 (1960), 6185.
(5)Harary, F.Graph Theory (Reading, Mass., Addison-Wesley, 1969).
(6)König, D.Theorie der endlichen und unendlichen Graphen (Leipzig 1936, reprinted Chelsea, New York, 1950).
(7)Mader, W.1-Faktoren von Graphen. Math. Ann. 201 (1973), 269282.
(8)Ore, O.Diameters in graphs. J. Combinatorial Theory 5 (1968), 7581.
(9)Thomassen, C.On hypohamiltonian graphs. Discrete Math. 10 (1974), 383390.
(10)Vizing, V. G.On an estimate of the chromatic class of a p-graph (Russian). Diskret. Analiz. 3 (1964), 2530.
(11)Vizing, V. G.The chromatic class of a multigraph. Cybernetics 1 (1965), 3241.
(12)Watkins, M. E. and Mesner, D. M.Cycles and connectivity in graphs. Can. J. Math. 19 (1967), 13191328.

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