# Colouring perfect planar graphs in parallel

## Abstract

Our algorithm assumes that the perfect planar graph is presented in the form of a planar drawing. However, this is not strictly necessary, for by [11] the problem of constructing a planar drawing of a planar graph is in NC; in particular, this problem can be solved in O(log^{2} n) time using O(n) processors on a CREW PRAM.

The original version of this algorithm bypassed the construction of the randomized algorithm, by using a different method of selecting a "nice" set of vertices at which to simultaneously perform α–β interchanges, as in the phase of interchanges in Stage 3. Here, a planar multigraph was constructed with vertices corresponding to the α–β components, with edges joining two vertices if and only if the corresponding α–β components had a common adjacent uncoloured vertex. This multigraph was then coloured using at most 5 colours and our "nice" set of vertices was chosen using this colouring. For more details, see [14]. We are indebted to an anonymous referee for pointing out Luby's technique and thus speeding up the algorithm considerably.

## Preview

Unable to display preview. Download preview PDF.

## References

- [1]K. APPEL AND W. HAKEN, Every planar map is four-colorable,
*Illinois J. Math.***21**(1977), 429–567.Google Scholar - [2]F. BAUERNÖPPEL AND H. JUNG, Fast parallel vertex coloring, in: Fundamentals of Computation Theory, Lecture Notes in Computer Science, Vol. 199, pp. 28–35 Springer, Berlin, 1985.Google Scholar
- [3]C. BERGE, Sur une conjecture relative au problème des codes optimaux,
*Comm. 13ieme Assemblee Gen. URSI*, Tokyo, 1962.Google Scholar - [4]J. BOYAR AND H. KARLOFF, Coloring planar graphs in parallel,
*J. Algorithms*,**8**, 470–479 (1987).Google Scholar - [5]K. DIKS, A fast parallel algorithm for six-colouring of planar graphs,
*12th Internat. Symp. on the Mathematical Foundations of Computer Science*, Bratislava, August, 1986.Google Scholar - [6]M. R. GAREY AND D.S. JOHNSON, "Computers and Intractability: A Guide to the Theory of NP-Completeness", Freeman, San Francisco, 1979.Google Scholar
- [7]F. HARARY, "Graph Theory", Addison-Wesley, Reading, MA, 1969.Google Scholar
- [8]W.-L. HSU, An O(n
^{3}) algorithm for the maximal independent set problem on planar perfect graphs,*J. ACM*, to appear.Google Scholar - [9]W.-L. HSU, Coloring planar perfect graphs by decomposition,
*Combinatorica*,**6**(4), (1986), p.381–385.Google Scholar - [10]W.-L. HSU, Recognizing planar perfect graphs,
*J. ACM*, Vol. 34, No.2, 1987, p. 255–288.Google Scholar - [11]P. N. KLEIN, J. H. REIF, An efficient parallel algorithm for planarity,
*Proc. 27th IEEE Annual Symposium on Foundations of Computer Science*, Toronto, 1986.Google Scholar - [12]M. LUBY, A simple parallel algorithm for the maximal independent set,
*SICOMP 15*(4) (1986).Google Scholar - [13]J. NAOR, A fast parallel coloring of planar graphs with five colors,
*Inf. Proc. Letters***25**(1987) 51–53.Google Scholar - [14]I. A. STEWART, A parallel algorithm to four-colour a perfect planar graph, Technical Report 245, Univ. of Newcastle upon Tyne, 1987.Google Scholar
- [15]I. A. STEWART, An Algorithm for Colouring Perfect Planar Graphs,
*Proc. 7th Conference on Foundations of Software Technology and Theoretical Computer Science*, Lecture Notes in Computer Science 287 (Ed. K.V. Nori), Springer-Verlag, Berlin, 1987, p.58–64.Google Scholar - [16]A. TUCKER, The strong perfect graph conjecture for planar graphs,
*Canad. J. Math.***25**(1973), 103–114.Google Scholar - [17]A. TUCKER AND D. WILSON, An O(N
^{2}) Algorithm for Coloring Perfect Planar Graphs,*J. Algorithms***5**, (1984) 60–68.Google Scholar