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Is there any polynomial upper bound for the universal labeling of graphs?

Ahadi, A ; Sharif University of Technology

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  1. Type of Document: Article
  2. DOI: 10.1007/s10878-016-0107-8
  3. Abstract:
  4. A universal labeling of a graph G is a labeling of the edge set in G such that in every orientation ℓ of G for every two adjacent vertices v and u, the sum of incoming edges of v and u in the oriented graph are different from each other. The universal labeling number of a graph G is the minimum number k such that G has universal labeling from { 1 , 2 , … , k} denoted it by χu→ (G). We have 2 Δ (G) - 2 ≤ χu→ (G) ≤ 2 Δ ( G ), where Δ (G) denotes the maximum degree of G. In this work, we offer a provocative question that is: “Is there any polynomial function f such that for every graph G, χu→ (G) ≤ f(Δ (G)) ?”. Towards this question, we introduce some lower and upper bounds on their parameter of interest. Also, we prove that for every tree T, χu→ (T) = O(Δ 3). Next, we show that for a given 3-regular graph G, the universal labeling number of G is 4 if and only if G belongs to Class 1. Therefore, for a given 3-regular graph G, it is an NP-complete to determine whether the universal labeling number of G is 4. Finally, using probabilistic methods, we almost confirm a weaker version of the problem. © 2016, Springer Science+Business Media New York
  5. Keywords:
  6. -+0741cgt ; Universal labeling ; Trees (mathematics) ; Adjacent vertices ; Lower and upper bounds ; Polynomial functions ; Probabilistic methods ; Regular graphs ; Trees ; Universal labeling number ; Graph theory
  7. Source: Journal of Combinatorial Optimization ; Volume 34, Issue 3 , 2017 , Pages 760-770 ; 13826905 (ISSN)
  8. URL: https://link.springer.com/article/10.1007/s10878-016-0107-8