Neighbor sum distinguishing index of -minor free graphs

A proper [ k ]-edge coloring of a graph G is a proper edge coloring of G using colors from . The neighbor set distinguishing index ndi ( G ) of G is the smallest integer k for which G admits a proper edge k -coloring such that any pair of adjacent vertices are incident to distinct sets of colors. A...

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Bibliographic Details
Published in:Graphs and combinatorics Vol. 41; no. 4; p. 90
Main Authors: Yang, Wei, Wu, Baoyindureng
Format: Journal Article
Language:English
Published: Tokyo Springer Japan 01.08.2025
Springer Nature B.V
Subjects:
Apexes
Applications of mathematics
Combinatorial analysis
Combinatorics
Engineering Design
Graph coloring
Graph theory
Graphs
Mathematics
Mathematics and Statistics
Original Paper
Sums
Neighbor set distinguishing index
Neighbor sum distinguishing index
minor free graphs
ISSN:0911-0119, 1435-5914
Online Access:Get full text
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Summary:A proper [ k ]-edge coloring of a graph G is a proper edge coloring of G using colors from . The neighbor set distinguishing index ndi ( G ) of G is the smallest integer k for which G admits a proper edge k -coloring such that any pair of adjacent vertices are incident to distinct sets of colors. A neighbor sum distinguishing [ k ]-edge coloring of G is a proper [ k ]-edge coloring of G such that for each edge , the sum of colors taken on the edges incident to u is different from the sum of colors taken on the edges incident to v . By nsdi ( G ), we denote the smallest value k in such a coloring of G . Apparently, for any graph G , . Wang and Wang (Applied Mathematics Letters 24 (2011) 2034-2037) proved that if G is a -minor free graph with , then . They posed an open problem: for a -minor free graph G with , is it ture that ? In addition, Zhang, Ding, Wang, Yan and Zhou (Graphs and combinatorics 32 (2016) 1621-1633) showed that if G is a -minor free graph with , then . They also proposed an open problem: let G be a -minor free graph with . Does it holds that ? In this paper, we show that the problems above with are true, improving a known result of Wang and Wang.
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ISSN:0911-0119
1435-5914
DOI:10.1007/s00373-025-02951-4