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Group connectivity under 3‐edge‐connectivity
Author(s) -
Han Miaomiao,
Li Jiaao,
Li Xueliang,
Wang Meiling
Publication year - 2021
Publication title -
journal of graph theory
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 54
eISSN - 1097-0118
pISSN - 0364-9024
DOI - 10.1002/jgt.22623
Subject(s) - mathematics , combinatorics , connectivity , abelian group , discrete mathematics , graph
Let S , T be two distinct finite Abelian groups with| S | = | T | . A fundamental theorem of Tutte shows that a graph admits a nowhere‐zero S ‐flow if and only if it admits a nowhere‐zero T ‐flow. Jaeger et al in 1992 introduced group connectivity as an extension of flow theory, and they asked whether such a relation holds for group connectivity analogy. It was negatively answered by Hušek et al in 2017 for graphs with edge‐connectivity 2 for the groups S = Z 4 and T = Z 2 2 . In this paper, we extend their results to 3‐edge‐connected graphs (including both cubic and general graphs), which answers open problems proposed by Hušek et al and Lai et al. Combining some previous results, this characterizes all the equivalence of group connectivity under 3‐edge‐connectivity, showing that every 3‐edge‐connected S ‐connected graph is T ‐connected if and only if{ S , T } ≠ { Z 4 , Z 2 2 } .