Abstract
Let G be a graph with a positive integer weight ω(v) for each vertex v. One wishes to assign each edge e of G a positive integer f(e) as a color so that ω(v) ≤ |f(e) − f(e′)| for any vertex v and any two edges e and e′ incident to v. Such an assignment f is called an ω-edge-coloring of G, and the maximum integer assigned to edges is called the span of f. The ω-chromatic index of G is the minimum span over all ω-edge-colorings of G. In the paper, we present various upper and lower bounds on the ω-chromatic index, and obtain three efficient algorithms to find an ω-edge-coloring of a given graph. One of them finds an ω-edge-coloring with span smaller than twice the ω-chromatic index.
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References
Cole, R., Ost, K., Schirra, S.: Edge-coloring bipartite multigraphs in \(\mathcal{{\it O}}(E\log D)\) time. Combinatorica 21(1), 5–12 (2001)
Corman, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms. MIT Press and McGraw Hill, Cambridge (2001)
Gabow, H.N., Nishizeki, T., Kariv, O., Leven, D., Terada, O.: Algorithms for edge-coloring graphs, Tech. Rept. TRECIS 41-85, Tohoku Univ. (1985)
Holyer, I.J.: The NP-completeness of edge coloring. SIAM J. on Computing 10, 718–721 (1981)
Jensen, T.R., Toft, B.: Graph Coloring Problems. John Wiley & Sons, New York (1995)
Karloff, H., Shmoys, D.B.: Efficient parallel algorithms for edge-coloring problems. J. of Algorithms 8(1), 39–52 (1987)
McDiamid, C.: On the span in channel assignment problems: bounds, computing and counting. Discrete Math 266, 387–397 (2003)
McDiamid, C., Reed, B.: Channel assignment on graphs of bounded treewidth. Discrete Math 273, 183–192 (2003)
Nakano, S., Nishizeki, T.: Edge-coloring problems for graphs. Interdisciplinary Information Sciences 1(1), 19–32 (1994)
Nishikawa, K., Nishizeki, T., Zhou, X.: Bandwidth consecutive multicolorings of graphs. Theoretical Computer Science 532, 64–72 (2014)
Obata, Y., Nishizeki, T.: Approximation Algorithms for Bandwidth Consecutive Multicolorings. In: Chen, J., Hopcroft, J.E., Wang, J. (eds.) FAW 2014. LNCS, vol. 8497, pp. 194–204. Springer, Heidelberg (2014)
Pinedo, M.L.: Scheduling: Theory. Springer Science, New York (2008)
Shannon, C.E.: A theorem on coloring the lines of a network. J. Math. Physics 28, 148–151 (1949)
Stiebitz, M., Scheide, D., Toft, B., Favrholdt, L.M.: Graph Edge Coloring. Wiley, Hoboken (2012)
West, D.B.: Introduction to Graph Theory. Prentice-Hall, Englewood Cliffs (1996)
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Obata, Y., Nishizeki, T. (2015). Edge-Colorings of Weighted Graphs. In: Rahman, M.S., Tomita, E. (eds) WALCOM: Algorithms and Computation. WALCOM 2015. Lecture Notes in Computer Science, vol 8973. Springer, Cham. https://doi.org/10.1007/978-3-319-15612-5_4
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DOI: https://doi.org/10.1007/978-3-319-15612-5_4
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