Graphs and Combinatorics

, Volume 29, Issue 5, pp 1329–1336 | Cite as

Neighbor Sum Distinguishing Index

  • Evelyne Flandrin
  • Antoni Marczyk
  • Jakub Przybyło
  • Jean-François Saclé
  • Mariusz Woźniak
Open Access
Original Paper

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.

Keywords

Proper edge coloring Neighbor-distinguishing index Neighbor sum distinguishing coloring Chromatic index 

Mathematics Subject Classification

05C15 

Notes

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.

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Copyright information

© The Author(s) 2012

Authors and Affiliations

  • Evelyne Flandrin
    • 1
  • Antoni Marczyk
    • 2
  • Jakub Przybyło
    • 2
  • Jean-François Saclé
    • 1
  • Mariusz Woźniak
    • 2
  1. 1.L R I, UMR 8623, Bât. 490Université de Paris-SudOrsayFrance
  2. 2.AGH University of Science and TechnologyKrakowPoland

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