[r, s, t]-Colorings of graphs

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Given non-negative integers r, s, and t, an [r, s, t]-coloring of a graph G = (V (G), E (G)) is a mapping c from V (G) ∪ E (G) to the color set { 0, 1, ..., k - 1 } such that | c (vi) - c (vj) | ≥ r for every two adjacent vertices vi, vj, | c (ei) - c (ej) | ≥ s for every two adjacent edges ei, ej, and | c (vi) - c (ej) | ≥ t for all pairs of incident vertices and edges, respectively. The [r, s, t]-chromatic numberχr, s, t (G) of G is defined to be the minimum k such that G admits an [r, s, t]-coloring. This is an obvious generalization of all classical graph colorings since c is a vertex coloring if r = 1, s = t = 0, an edge coloring if s = 1, r = t = 0, and a total coloring if r = s = t = 1, respectively. We present first results on χr, s, t (G) such as general bounds and also exact values, for example if min { r, s, t } = 0 or if G is a complete graph. © 2006 Elsevier B.V. All rights reserved.




Kemnitz, A., & Marangio, M. (2007). [r, s, t]-Colorings of graphs. Discrete Mathematics, 307(2), 199–207. https://doi.org/10.1016/j.disc.2006.06.030

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