# 4-chromatic Koester graphs

Andrey A. Dobrynin; Leonid S. Mel'nikov

Discussiones Mathematicae Graph Theory (2012)

- Volume: 32, Issue: 4, page 617-627
- ISSN: 2083-5892

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topAndrey A. Dobrynin, and Leonid S. Mel'nikov. "4-chromatic Koester graphs." Discussiones Mathematicae Graph Theory 32.4 (2012): 617-627. <http://eudml.org/doc/271063>.

@article{AndreyA2012,

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.},

author = {Andrey A. Dobrynin, Leonid S. Mel'nikov},

journal = {Discussiones Mathematicae Graph Theory},

keywords = {planar graph; 4-critical graph; Grötzsch-Sachs graph; Koester graph},

language = {eng},

number = {4},

pages = {617-627},

title = {4-chromatic Koester graphs},

url = {http://eudml.org/doc/271063},

volume = {32},

year = {2012},

}

TY - JOUR

AU - Andrey A. Dobrynin

AU - Leonid S. Mel'nikov

TI - 4-chromatic Koester graphs

JO - Discussiones Mathematicae Graph Theory

PY - 2012

VL - 32

IS - 4

SP - 617

EP - 627

AB - 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.

LA - eng

KW - planar graph; 4-critical graph; Grötzsch-Sachs graph; Koester graph

UR - http://eudml.org/doc/271063

ER -

## References

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- [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. Zbl1299.05121
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- [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).
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