Discussiones Mathematicae Graph Theory 30(4) (2010) 619-636
doi: 10.7151/dmgt.1518

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VERTEX COLORING THE SQUARE OF OUTERPLANAR GRAPHS OF LOW DEGREE

Geir Agnarsson

Department of Mathematical Sciences
George Mason University
MS 3F2, 4400 University Drive, Fairfax, VA 22030, USA
e-mail: geir@math.gmu.edu

Magnús M. Halldórsson

Department of Computer Science
Reykjavík University
IS-103 Reykjavík, Iceland
e-mail: mmh@ru.is

Abstract

Vertex colorings of the square of an outerplanar graph have received a lot of attention recently. In this article we prove that the chromatic number of the square of an outerplanar graph of maximum degree Δ = 6 is 7. The optimal upper bound for the chromatic number of the square of an outerplanar graph of maximum degree Δ ≠ 6 is known. Hence, this mentioned chromatic number of 7 is the last and only unknown upper bound of the chromatic number in terms of Δ.

Keywords: outerplanar, chromatic number, power of a graph, weak dual.

2010 Mathematics Subject Classification: Primary: 05C05,
05C15; Secondary: 05C10.

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Received 31 July 2009
Revised 11 January 2010
Accepted 11 January 2010