ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2018-2019): c. 84%

Discussiones Mathematicae Graph Theory


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


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: (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.


[1] R.L. Brooks, On coloring the nodes of a network, Proc. Cambridge Phil. Soc. 37 (1941) 194-197, doi: 10.1017/S030500410002168X.
[2] Chao Chong-Yun, A critically chromatic graph, Discrete Math. 172 (1997) 3-7, doi: 10.1016/S0012-365X(96)00262-2.
[3] G.A. Dirac, 4-chrome Graphen Trennende und vollständige 4-Graphen, Math. Nachr. 22 (1960) 51-60, doi: 10.1002/mana.19600220106.
[4] G.A. Dirac, In abstrakten Graphen vorhandene vollständige 4-Graphen und ihre Unterteilungen, Math. Nachr. 22 (1960) 61-85, doi: 10.1002/mana.19600220107.
[5] A.A. Dobrynin, L.S. Mel'nikov and A.V. Pyatkin, 4-chromatic edge-critical regular graphs with high connectivity, Proc. Russian Conf. Discrete Analysis and Operation Research (DAOR-2002), Novosibirsk, pp. 25-30 (in Russian).
[6] A.A. Dobrynin, L.S. Mel'nikov and A.V. Pyatkin, On 4-chromatic edge-critical regular graphs of high connectivity, Discrete Math. 260 (2003) 315-319, doi: 10.1016/S0012-365X(02)00668-4.
[7] A.A. Dobrynin, L.S. Mel'nikov and A.V. Pyatkin, Regular 4-critical graphs of even degree, J. Graph Theory 46 (2004) 103-130, doi: 10.1002/jgt.10176.
[8] P. Erdös, On some aspects of my work with Gabriel Dirac, in: L.D. Andersen, I.T. Jakobsen, C. Thomassen, B. Toft and P.D. Vestergaard (Eds.), Graph Theory in Memory of G.A. Dirac, Annals of Discrete Mathematics, Vol. 41, North-Holland, 1989, pp. 111-116.
[9] V.A. Evstigneev and L.S. Mel'nikov, Problems and Exercises on Graph Theory and Combinatorics (Novosibirsk State University, Novosibirsk, 1981) (in Russian).
[10] T. Gallai, Kritische Graphen I., Publ. Math. Inst. Hungar. Acad. Sci. 8 (1963) 165-192.
[11] M.R. Garey and D.S. Johnson, Computers and Intractability. A Guide to the Theory of NP-Completeness (W.H. Freeman and Company, San Francisco, 1979).
[12] F. Göbel and E.A. Neutel, Cyclic graphs, Discrete Appl. Math. 99 (2000) 3-12, doi: 10.1016/S0166-218X(99)00121-3.
[13] T.R. Jensen, Dense critical and vertex-critical graphs, Discrete Math. 258 (2002) 63-84, doi: 10.1016/S0012-365X(02)00262-5.
[14] T.R. Jensen and G.F. Royle, Small graphs of chromatic number 5: a computer search, J. Graph Theory 19 (1995) 107-116, doi: 10.1002/jgt.3190190111.
[15] T.R. Jensen and B. Toft, Graph Coloring Problems (John Wiley & Sons, USA, 1995).
[16] G. Koester, Note to a problem of T. Gallai and G.A. Dirac, Combinatorica 5 (1985) 227-228, doi: 10.1007/BF02579365.
[17] G. Koester, 4-critical 4-valent planar graphs constructed with crowns, Math. Scand. 67 (1990) 15-22.
[18] G. Koester, On 4-critical planar graphs with high edge density, Discrete Math. 98 (1991) 147-151, doi: 10.1016/0012-365X(91)90039-5.
[19] W. Mader, Über den Zusammenhang symmetrischer Graphen, Arch. Math. (Basel) 21 (1970) 331-336, doi: 10.1007/BF01220924.
[20] W. Mader, Eine Eigenschaft der Atome endlicher Graphen, Arch. Math. (Basel) 22 (1971) 333-336, doi: 10.1007/BF01222585.
[21] A.V. Pyatkin, 6-regular 4-critical graph, J. Graph Theory 41 (2002) 286-291, doi: 10.1002/jgt.10066.
[22] M.E. Watkins, Some classes of hypoconnected vertex-transitive graphs, in: Recent Progress in Combinatorics (Academic Press, New-York, 1969) 323-328.
[23] M.E. Watkins, Connectivity of transitive graphs, J. Combin. Theory 8 (1970) 23-29, doi: 10.1016/S0021-9800(70)80005-9.
[24] D.A. Youngs, Gallai's problem on Dirac's construction, Discrete Math. 101 (1992) 343-350, doi: 10.1016/0012-365X(92)90615-M.

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