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.
The first author has been supported by EPSRC under project EP/G043434/1. The second and third author have been supported by ANR Blanc AGAPE (ANR-09-BLAN-0159-03).
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Golovach, P.A., Kratsch, D., Couturier, JF. (2010). Colorings with Few Colors: Counting, Enumeration and Combinatorial Bounds. In: Thilikos, D.M. (eds) Graph Theoretic Concepts in Computer Science. WG 2010. Lecture Notes in Computer Science, vol 6410. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-16926-7_6
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