Discussiones Mathematicae Graph Theory 27(2) (2007) 269-279
doi: 10.7151/dmgt.1360

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Andrey A. Dobrynin,  Leonid S. Mel'nikov and Artem V. Pyatkin

Sobolev Institute of Mathematics
Siberian Branch, Russian Academy of Sciences
Novosibirsk 630090, Russia
e-mail: dobr@math.nsc.ru (A.A. Dobrynin)


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.

Keywords: vertex coloring, 4-critical graph, circulant, regular graph, vertex connectivity.

2000 Mathematics Subject Classification: 05C15.


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Received 9 February 2006
Revised 28 February 2007
Accepted 12 March 2007