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2007 | 27 | 2 | 269-279
Tytuł artykułu

Erdős regular graphs of even degree

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Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Rocznik
Tom
27
Numer
2
Strony
269-279
Opis fizyczny
Daty
wydano
2007
otrzymano
2006-02-09
poprawiono
2007-02-28
zaakceptowano
2007-03-12
Twórcy
  • Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk 630090, Russia
  • Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk 630090, Russia
  • Sobolev Institute of Mathematics, Siberian Branch, Russian Academy of Sciences, Novosibirsk 630090, Russia
Bibliografia
  • [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.
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.bwnjournal-article-doi-10_7151_dmgt_1360
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