Abstract
An acyclic edge-colouring of a graph is a proper edge-colouring such that the subgraph induced by the edges of any two colours is acyclic. The acyclic chromatic index of a graph G is the smallest possible number of colours in an acyclic edge-colouring of G. In [12], we have shown that the acyclic chromatic index of a connected subcubic graph G is at most 4 with the exception of K 4 and K 3,3, for which five colors are optimal. Here we give a quadratic-time algorithm that finds an acyclic 4-edge-colouring of a given connected subcubic graph different from K 4 and K 3,3.
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References
Alon, N., Sudakov, B., Zaks, A.: Acyclic edge-colorings of graphs. J. Graph Theory 37, 157–167 (2001)
Alon, N., McDiarmid, C.J.H., Reed, B.A.: Acyclic coloring of graphs. Random Struct. Algorithms 2, 277–288 (1991)
Alon, N., Zaks, A.: Algorithmic aspects of acyclic edge colorings. Algorithmica 32, 611–614 (2002)
Basavaraju, M., Chandran, L.S.: Acyclic edge coloring of subcubic graphs. Discrete Math. (in press)
Burstein, M.I.: Every 4-valent graph has an acyclic 5-coloring (in Russian). Soobšč. Akad. Nauk Gruzin. SSR 93, 21–24 (1979) (Georgian and English summaries)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 2nd edn., pp. 558–559. The MIT Press, Cambridge (2001)
Fiamčík, J.: Acyclic chromatic index of a graph with maximum valency three. Arch. Math. (Brno) 16, 81–88 (1980)
Fiamčík, J.: Acyclic chromatic index of subdivided graph. Arch. Math. (Brno) 20, 69–82 (1984)
Fiamčík, J.: The acyclic chromatic class of a graph (in russian). Math. Slovaca 28, 139–145 (1978)
Gebremedhin, A.H., Tarafdar, A., Manne, F., Pothen, A.: New acyclic and star coloring algorithms with application to computing Hessians. SIAM J. Sci. Comput. 29(3), 1042–1072 (2007)
Grünbaum, B.: Acyclic colorings of planar graphs. Israel Journal of Mathematics, 390–408 (1973)
Máčajová, E., Mazák, J.: Optimal acyclic edge-colouring of cubic graphs, technical report (2008), http://kedrigern.dcs.fmph.uniba.sk/reports/download.php?id=18
Molloy, M., Reed, B.: Further algorithmic aspects of the local lemma. In: Proceedings of the 30th Annual ACM Symposium on the Theory of Computing, May 1998, pp. 524–529 (May 1998)
Nešetřil, J., Wormald, N.C.: The acyclic edge chromatic number of a random d-regular graph is d + 1. J. Graph Theory 49, 69–74 (2005)
Skulrattanakulchai, S.: Acyclic colouring of subcubic graphs. Information Processing Letters 92, 161–167 (2004)
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Máčajová, E., Mazák, J. (2011). An Algorithm for Optimal Acyclic Edge-Colouring of Cubic Graphs. In: Atallah, M., Li, XY., Zhu, B. (eds) Frontiers in Algorithmics and Algorithmic Aspects in Information and Management. Lecture Notes in Computer Science, vol 6681. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21204-8_17
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DOI: https://doi.org/10.1007/978-3-642-21204-8_17
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