Abstract
Edge coloring, total coloring and L(2,1)-labeling are well-studied NP-hard graph problems. Even the versions asking whether a graph has a coloring with few colors or a labeling with few labels remain NP-hard on graphs of small maximum degree.
This paper studies enumeration and counting problems on edge colorings, total colorings and L(2,1)-labelings of graphs. One part deals with the enumeration of all edge 3-colorings, all total 4-colorings and all L(2,1)-labelings of span 5 of a given connected cubic graph. Branching algorithms to solve these enumeration problems are established. They imply upper bounds on the maximum number of edge 3-colorings, total 4-colorings and L(2,1)-labelings of span 5 in any n-vertex connected cubic graphs. Corresponding combinatorial lower bounds are also provided.
The other part of the paper studies dynamic programming algorithms solving counting problems. On one hand, algorithms to count the number of edge k-colorings and total k-colorings for graphs of bounded pathwidth are given. On the other hand, an algorithm to count the number of L(2,1)-labelings of span 4 for graphs of maximum degree three are given.
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Notes
For the purpose of this paper we shall address L(2,1)-labeling as a coloring problem.
As has recently become standard, we write f(n)=O ∗(g(n)) if f(n)≤p(n)⋅g(n) for some polynomial p(n).
Motivated by a presentation of the edge coloring results of our work, published in [18], in a talk given by D. Kratsch at a meeting of the AGAPE project in January 2011, S. Bessy and F. Havet first (during the meeting) established the maximum number of edge 3-colorings in cubic graphs, and then extended this in various directions [4]. Let us mention that their work improves upon our enumeration algorithms for edge 3-colorings and total 4-colorings of connected cubic graphs and the corresponding combinatorial upper bounds given in Sect. 3; they study neither the L(2,1)-labeling problem nor counting versions of the problems.
The auxiliary graph H is the only multigraph of the paper and its only purpose is to ease the description of the example.
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Acknowledgement
We would like to thank anonymous referees for their careful reading of an earlier version and the helpful comments.
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An extended abstract of this paper was presented at the 36th International Workshop on Graph Theoretic Concepts in Computer Science, Zarós, Crete, Greece, June 28–30, 2010 and published in the Proceedings of WG 2010 [18].
J.-F. Couturier, D. Kratsch and M. Liedloff were supported by ANR under project AGAPE (ANR-09-BLAN-0159-03).
P.A. Golovach was supported by EPSRC under project EP/G043434/1.
A. Pyatkin was supported by EPSRC Grant EP/F064551/1, by the RFBR projects 12-01-00184-a and 12-01-00093-a, and by the Ministry of education and science of the Russian Federation (contract number 14.740.11.0868).
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Couturier, JF., Golovach, P.A., Kratsch, D. et al. Colorings with few Colors: Counting, Enumeration and Combinatorial Bounds. Theory Comput Syst 52, 645–667 (2013). https://doi.org/10.1007/s00224-012-9410-7
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DOI: https://doi.org/10.1007/s00224-012-9410-7