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r-Strong edge colorings of graphs

Akbari, S ; Sharif University of Technology | 2006

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  1. Type of Document: Article
  2. DOI: 10.1016/j.disc.2004.12.027
  3. Publisher: Elsevier , 2006
  4. Abstract:
  5. Let G be a graph and for any natural number r, χs ′ (G, r) denotes the minimum number of colors required for a proper edge coloring of G in which no two vertices with distance at most r are incident to edges colored with the same set of colors. In [Z. Zhang, L. Liu, J. Wang, Adjacent strong edge coloring of graphs, Appl. Math. Lett. 15 (2002) 623-626] it has been proved that for any tree T with at least three vertices, χs ′ (T, 1) ≤ Δ (T) + 1. Here we generalize this result and show that χs ′ (T, 2) ≤ Δ (T) + 1. Moreover, we show that if for any two vertices u and v with maximum degree d (u, v) ≥ 3, then χs ′ (T, 2) = Δ (T). Also for any tree T with Δ (T) ≥ 3 we prove that χs ′ (T, 3) ≤ 2 Δ (T) - 1. Finally, it is shown that for any graph G with no isolated edges, χs ′ (G, 1) ≤ 3 Δ (G). © 2006 Elsevier B.V. All rights reserved
  6. Keywords:
  7. Color ; Coloring ; Number theory ; Set theory ; Trees (mathematics) ; Graph coloring ; Natural numbers ; Strong edge coloring ; Graph theory
  8. Source: Discrete Mathematics ; Volume 306, Issue 23 SPEC. ISS , 2006 , Pages 3005-3010 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0012365X06003840