On Brooks' theorem and some related results.
H. Tverberg (1983)
Mathematica Scandinavica
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H. Tverberg (1983)
Mathematica Scandinavica
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Gabriel Dirac (1972)
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F. Harary, G. Exoo, J. Kabell (1981)
Mathematica Scandinavica
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Lars Dovling Andersen (1977)
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Ivan Tafteberg Jakobsen (1972)
Mathematica Scandinavica
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G.A. Dirac (1963)
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Ivan Tafteberg Jakobsen (1983)
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Tudor Zamfirescu (1976)
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Monika Schmidt-Steup (1986)
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Masaki Izumi (1993)
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R. Ayala, E. Dominguez, A. Márquez (1995)
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Lars Dovling Andersen (1989)
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Lucas C. van der Merwe, Cristine M. Mynhardt, Teresa W. Haynes (2001)
Discussiones Mathematicae Graph Theory
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Denote the total domination number of a graph G by γₜ(G). A graph G is said to be total domination edge critical, or simply γₜ-critical, if γₜ(G+e) < γₜ(G) for each edge e ∈ E(G̅). For 3ₜ-critical graphs G, that is, γₜ-critical graphs with γₜ(G) = 3, the diameter of G is either 2 or 3. We characterise the 3ₜ-critical graphs G with diam G = 3.
Andrey A. Dobrynin, Leonid S. Mel&#039;nikov, Artem V. Pyatkin (2007)
Discussiones Mathematicae Graph Theory
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In 1960, Dirac put forward the conjecture that r-connected 4-critical graphs exist for every r ≥ 3. In 1989, Erdös conjectured that for every r ≥ 3 there exist r-regular 4-critical graphs. A method for finding r-regular 4-critical graphs and the numbers of such graphs for r ≤ 10 have been reported in [6,7]. Results of a computer search for graphs of degree r = 12,14,16 are presented. All the graphs found are both r-regular and r-connected.