Abstract.
We generalize the concept of a 2-coloring of a graph to what we call a semi-balanced coloring by relaxing a certain discrepancy condition on the shortest-paths hypergraph of the graph. Let G be an undirected, unweighted, connected graph with n vertices and m edges. We prove that the number of different semi-balanced colorings of G is: (1) at most n+1 if G is bipartite; (2) at most m if G is non-bipartite and triangle-free; and (3) at most m+1 if G is non-bipartite. Based on the above combinatorial investigation, we design an algorithm to enumerate all semi-balanced colorings of G in O(n m 2) time.
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Acknowledgments The authors thank Tetsuo Asano, Naoki Katoh, Kunihiko Sadakane, and Hisao Tamaki for helpful discussions and comments.
Supported in part by Sweden-Japan Foundation
Final version received: November 17, 2003
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Jansson, J., Tokuyama, T. Semi-Balanced Colorings of Graphs: Generalized 2-Colorings Based on a Relaxed Discrepancy Condition. Graphs and Combinatorics 20, 205–222 (2004). https://doi.org/10.1007/s00373-004-0557-0
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DOI: https://doi.org/10.1007/s00373-004-0557-0