Abstract
For a proper edge coloring of a graph G the palette S(v) of a vertex v is the set of the colors of the incident edges. If S(u) ≠ S(v) then the two vertices u and v of G are distinguished by the coloring. A d-strong edge coloring of G is a proper edge coloring that distinguishes all pairs of vertices u and v with distance 1 ≤ d (u, v) ≤ d. The d-strong chromatic index \({\chi_{d}^{\prime}(G)}\) of G is the minimum number of colors of a d-strong edge coloring of G. Such colorings generalize strong edge colorings and adjacent strong edge colorings as well. We prove some general bounds for \({\chi_{d}^{\prime}(G)}\) , determine \({\chi_{d}^{\prime}(G)}\) completely for paths and give exact values for cycles disproving a general conjecture of Zhang et al. (Acta Math Sinica Chin Ser 49:703–708 2006)).
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Kemnitz, A., Marangio, M. d-strong Edge Colorings of Graphs. Graphs and Combinatorics 30, 183–195 (2014). https://doi.org/10.1007/s00373-012-1251-2
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DOI: https://doi.org/10.1007/s00373-012-1251-2