Abstract
We consider proper edge colorings of a graph G using colors of the set {1, . . . , k}. Such a coloring is called neighbor sum distinguishing if for any pair of adjacent vertices x and y the sum of colors taken on the edges incident to x is different from the sum of colors taken on the edges incident to y. The smallest value of k in such a coloring of G is denoted by ndiΣ(G). In the paper we conjecture that for any connected graph G ≠ C 5 of order n ≥ 3 we have ndiΣ(G) ≤ Δ(G) + 2. We prove this conjecture for several classes of graphs. We also show that ndiΣ(G) ≤ 7Δ(G)/2 for any graph G with Δ(G) ≥ 2 and ndiΣ(G) ≤ 8 if G is cubic.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Akbari S., Bidkhori H., Nosrati N.: r-Strong edge colorings of graphs. Discrete. Math. 306, 3005–3010 (2006)
Bondy J.A., Murty U.S.R.: Graph Theory with Applications. Macmillan, London (1976)
Balister PN., Győri E., Lehel L., Schelp R.H.: Adjacent vertex distinguishing edge-colorings. SIAM. J. Discrete. Math. 21, 237–250 (2007)
Dénes, J., Keedwell, A.D.: Transversal and Complete Mappings, In: Latin Squares. New Developments in the Theory and Applications, Elsevier, Amsterdam (1991)
Edwards K., Horňák M., Woźniak M.: On the neighbour-distinguishing index of a graph. Graphs. Combin. 22, 341–350 (2006)
Hatami H.: is a bound on the adjacent vertex distinguishing edge chromatic number. J. Combin. Theory. Ser B. 95, 246–256 (2005)
Mahéo, M., Saclé, J-F.: Some results on (Σ, p, g)-valuation of connected graphs, Rapport de Recherche 1497, Université de Paris-Sud, Centre d’Orsay (2008)
Zhang Z., Liu L., Wang J.: Adjacent strong edge coloring of graph. Appl. Math. Lett. 15, 623–626 (2005)
Open Access
This article is distributed under the terms of the Creative Commons Attribution License which permits any use, distribution, and reproduction in any medium, provided the original author(s) and the source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
The work of the authors from AGH University of Science and Technology was partially supported by the Polish Ministry of Science and Higher Education.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Flandrin, E., Marczyk, A., Przybyło, J. et al. Neighbor Sum Distinguishing Index. Graphs and Combinatorics 29, 1329–1336 (2013). https://doi.org/10.1007/s00373-012-1191-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00373-012-1191-x
Keywords
- Proper edge coloring
- Neighbor-distinguishing index
- Neighbor sum distinguishing coloring
- Chromatic index