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Minimal coloring and strength of graphs

Hajiabolhassan, H ; Sharif University of Technology | 2000

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  1. Type of Document: Article
  2. DOI: 10.1016/S0012-365X(99)00319-2
  3. Publisher: Elsevier , 2000
  4. Abstract:
  5. Let G be a graph. A minimal coloring of G is a coloring which has the smallest possible sum among all proper colorings of G, using natural numbers. The vertex-strength of G, denoted by s(G), is the minimum number of colors which is necessary to obtain a minimal coloring. In this note we study these concepts, and define a new concept called the edge-strength of G, denoted by s′(G). We prove the celebrated Brooks' theorem for χ(G) replaced by s(G) and we also prove the following upper bound for s(G): s(G) ≤ ⌈col(G) + Δ(G)/2⌉, where col(G) is an invariant based on linear orderings of the vertices. Also, it is proved that s′(G) lies between Δ(G) and Δ(G) + 1, as for χ′(G), but it may be not equal to Δ′(G). Based on our results about vertex-strength we conjecture s(G) ≤ ⌈χ(G) + Δ(G)/2⌉. © 2000 Published by Elsevier Science B.V. All rights reserved
  6. Keywords:
  7. Chromatic sum ; Minimal coloring ; Strength
  8. Source: Discrete Mathematics ; Volume 215, Issue 1-3 , 2000 , Pages 265-270 ; 0012365X (ISSN)
  9. URL: https://www.sciencedirect.com/science/article/pii/S0012365X99003192