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On Group Connectivity of Graphs

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Abstract

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero Z 3-flow and Jaeger et al. [Group connectivity of graphs–a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165-182] further conjectured that every 5-edge-connected graph is Z 3-connected. These two conjectures are in general open and few results are known so far. A weaker version of Tutte’s conjecture states that every 4-edge-connected graph with each edge contained in a circuit of length at most 3 admits a nowhere-zero Z 3-flow. Devos proposed a stronger version problem by asking if every such graph is Z 3-connected. In this paper, we first answer this later question in negative and get an infinite family of such graphs which are not Z 3-connected. Moreover, motivated by these graphs, we prove that every 6-edge-connected graph whose edge set is an edge disjoint union of circuits of length at most 3 is Z 3-connected. It is a partial result to Jaeger’s Z 3-connectivity conjecture.

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Authors and Affiliations

  1. Department of Mathematics, West Virginia University, Morgantown, WV, 26506, USA

    Hong-Jian Lai & Ju Zhou

  2. Department of Mathematics, University if West Georgia, Carrollton, GA, 30118, USA

    Rui Xu

Authors
  1. Hong-Jian Lai
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  2. Rui Xu
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  3. Ju Zhou
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Correspondence to Rui Xu.

Additional information

Received: May 23, 2006. Final version received: January 13, 2008

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Lai, HJ., Xu, R. & Zhou, J. On Group Connectivity of Graphs. Graphs and Combinatorics 24, 195–203 (2008). https://doi.org/10.1007/s00373-008-0780-1

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  • DOI: https://doi.org/10.1007/s00373-008-0780-1

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