Colorings with few colors: counting, enumeration and combinatorial bounds

Authors:
Petr A. Golovach;Dieter Kratsch;Jean-Francois Couturier
Affiliations:
School of Engineering and Computing Sciences, Durham University, Durham, United Kingdom;Laboratoire d'Informatique Théorique et Appliquée, Université Paul Verlaine-Metz, Metz Cedex 01, France;Laboratoire d'Informatique Théorique et Appliquée, Université Paul Verlaine-Metz, Metz Cedex 01, France
Venue:
WG'10 Proceedings of the 36th international conference on Graph-theoretic concepts in computer science
Year:
2010

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Abstract

We provide exact algorithms for enumeration and counting problems on edge colorings and total colorings of graphs, if the number of (available) colors is fixed and small. For edge 3-colorings the following is achieved: there is a branching algorithm to enumerate all edge 3- colorings of a connected cubic graph in time O*(25n/8). This implies that the maximum number of edge 3-colorings in an n-vertex connected cubic graph is O*(25n/8). Finally, the maximum number of edge 3-colorings in an n-vertex connected cubic graph is lower bounded by 12n/10. Similar results are achieved for total 4-colorings of connected cubic graphs. We also present dynamic programming algorithms to count the number of edge k-colorings and total k-colorings for graphs of bounded pathwidth. These algorithms can be used to obtain fast exact exponential time algorithms for counting edge k-colorings and total k-colorings on graphs, if k is small.