Discussiones Mathematicae Graph Theory 32(4) (2012) 617-627
doi: 10.7151/dmgt.1630

[BIBTex] [PDF] [PS]

4-chromatic Koester Graphs

Andrey A. Dobrynin and Leonid S. Mel'nikov

Sobolev Institute of Mathematics
Siberian Branch of the Russian Academy of Sciences
Novosibirsk 630090, Russia

Abstract

Let G be a simple 4-regular plane graph and let S be a decomposition of G into edge-disjoint cycles. Suppose that every two adjacent edges on a face belong to different cycles of S. Such a graph G arises as a superposition of simple closed curves in the plane with tangencies disallowed. Studies of coloring of graphs of this kind were originated by Grötzsch. Two 4-chromatic graphs generated by circles in the plane were constructed by Koester in 1984 [10,11,12]. Until now, no other examples of such graphs were known. We present fourteen new 4-chromatic graphs generated by circles in the plane.

Keywords: planar graph, 4-critical graph, Grötzsch-Sachs graph, Koester graph

2010 Mathematics Subject Classification: 05C10, 05C15.

References

[1]A.A. Dobrynin and L.S. Mel'nikov, Counterexamples to Grötzsch-Sachs-Koester's conjecture, Discrete Math. 306 (2006) 591--594, doi: 10.1016/j.disc.2005.08.010.
[2]A.A. Dobrynin and L.S. Mel'nikov, Two series of edge 4-critical Grötzsch-Sachs graphs generated by four curves in the plane, Siberian Electronic Math. Reports 5 (2008) 255--278.
[3]A.A. Dobrynin and L.S. Mel'nikov, Infinite families of 4-chromatic Grötzsch-Sachs graphs, J. Graph Theory 59 (2008) 279--292, doi: 10.1002/jgt.20339.
[4]A.A. Dobrynin and L.S. Mel'nikov, 4-chromatic edge critical Grötzsch-Sachs graphs, Discrete Math. 309 (2009) 2564--2566, doi: 10.1016/j.disc.2008.06.006.
[5]H. Sachs (Ed.), Graphs, Hypergraphs and Applications, Proc. Conference on Graph Theory, Eyba, 1984, (B.G. Teubner Verlagsgesellschaft, 1985).
[6]F. Jaeger, On nowhere-zero flows in multigraphs, {Proc. Fifth British Combinatorial Conference 1975}, C.St.J.A. Nash-Williams and J.Sheehan (Ed(s)), (Congressus Numerantium XV, Winnipeg, Utilitas Mathematica Publising, Inc. 1976) 373--378.
[7]F. Jaeger, Sur les graphes couverts par leurs bicycles et la conjecture des quatre couleurs, in: Probl`emes Combinatoires et Theorie des Graphes, J.-C. Bermond, J.- C. Fournier, M. Las Vergnas and D. Sotteau (Ed(s)), (Paris, Editions du Centre National de la Recherche Scientifique, 1978) 243?247.
[8]F. Jaeger and H. Sachs, Problem, in: Graph Theory in memory of G.A. Dirac, ed(s), L.D. Andersen, I.T. Jakobsen, C. Thomassen, B. Toft, P.D. Vestergaard Ann. Discrete Math. 41, 1989) 515.
[9]T. Jensen and B. Toft, Graph Coloring Problems (John Wiley & Sons, New York, 1995).
[10]G. Koester, Bemerkung zu einem Problem von H. Grötzsch, Wiss. Z. Univ. Halle 33 (1984) 129.
[11]G. Koester, Coloring problems on a class of 4-regular planar graphs, in: Graphs, Hypergraphs and Applications. Proc. Conference on Graph Theory, Eyba, 1984, ed(s), H. Sachs B.G. Teubner Verlagsgesellschaft, 1985) 102--105.
[12]G. Koester, Note to a problem of T. Gallai and G. A. Dirac, Combinatorica 5 (1985) 227--228, doi: 10.1007/BF02579365.
[13]L.S. Mel?nikov, A.A. Dobrynin and G. Koester, 4-chromatic Grötzsch-Sachs graphs and edge 4-critical 4-valent planar graphs, some remarks on older and latest results, Report on Conf. Graph Theory on the Occasion of the 80th Birthday of Prof. Horst Sachs (Technical University Ilmenau, Germany, Ilmenau, March, 2007).
[14]H. Sachs, Problem, Math. Balkanica 4 (1974) 536.
[15]H. H. Sachs, A Three-Colour-Conjecture of Grötzsch, in: Probl`emes Combinatoires et Theorie des Graphes, J.-C. Bermond, J.-C. Fournier, M. Las Vergnas and D. Sotteau (Ed(s)), (Paris, Editions du Centre National de la Recherche Scientifique, 1978) 441.
[16]R. Steinberg, The state of the three color problem, in: Quo Vadis, Graph Theory?, ed(s), J. Gimbel, J.W. Kennedy, L.V. Quintas Annals Discrete Math. 55, 1993) 211--248.

Received 26 July 2011
Revised 15 November 2011
Accepted 18 November 2011