# 2-Distance vertex-distinguishing index of subcubic graphs

2-Distance vertex-distinguishing index of subcubic graphs A 2-distance vertex-distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets of colors. The 2-distance vertex-distinguishing index $$\chi ^{\prime }_{\mathrm{d2}}(G)$$ χ d 2 ′ ( G ) of G is the minimum number of colors needed for a 2-distance vertex-distinguishing edge coloring of G. Some network problems can be converted to the 2-distance vertex-distinguishing edge coloring of graphs. It is proved in this paper that if G is a subcubic graph, then $$\chi ^{\prime }_{\mathrm{d2}}(G)\le 6$$ χ d 2 ′ ( G ) ≤ 6 . Since the Peterson graph P satisfies $$\chi ^{\prime }_{\mathrm{d2}}(P)=5$$ χ d 2 ′ ( P ) = 5 , our solution is within one color from optimal. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Combinatorial Optimization Springer Journals

# 2-Distance vertex-distinguishing index of subcubic graphs

, Volume 36 (1) – Apr 6, 2018
13 pages

/lp/springer_journal/2-distance-vertex-distinguishing-index-of-subcubic-graphs-toLamWMYtt
Publisher
Springer US
Subject
Mathematics; Combinatorics; Convex and Discrete Geometry; Mathematical Modeling and Industrial Mathematics; Theory of Computation; Optimization; Operations Research/Decision Theory
ISSN
1382-6905
eISSN
1573-2886
D.O.I.
10.1007/s10878-018-0288-4
Publisher site
See Article on Publisher Site

### Abstract

A 2-distance vertex-distinguishing edge coloring of a graph G is a proper edge coloring of G such that any pair of vertices at distance 2 have distinct sets of colors. The 2-distance vertex-distinguishing index $$\chi ^{\prime }_{\mathrm{d2}}(G)$$ χ d 2 ′ ( G ) of G is the minimum number of colors needed for a 2-distance vertex-distinguishing edge coloring of G. Some network problems can be converted to the 2-distance vertex-distinguishing edge coloring of graphs. It is proved in this paper that if G is a subcubic graph, then $$\chi ^{\prime }_{\mathrm{d2}}(G)\le 6$$ χ d 2 ′ ( G ) ≤ 6 . Since the Peterson graph P satisfies $$\chi ^{\prime }_{\mathrm{d2}}(P)=5$$ χ d 2 ′ ( P ) = 5 , our solution is within one color from optimal.

### Journal

Journal of Combinatorial OptimizationSpringer Journals

Published: Apr 6, 2018

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