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All 4-Edge-Connected HHD-Free Graphs are \({\mathbb{Z}_3}\)-Connected

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Abstract

An undirected graph G = (V, E) is called \({\mathbb{Z}_3}\)-connected if for all \({b: V \rightarrow \mathbb{Z}_3}\) with \({\sum_{v \in V}b(v)=0}\), an orientation D = (V, A) of G has a \({\mathbb{Z}_3}\)-valued nowhere-zero flow \({f: A\rightarrow \mathbb{Z}_3-\{0\}}\) such that \({\sum_{e \in \delta^+(v)}f(e)-\sum_{e \in \delta^-(v)}f(e)=b(v)}\) for all \({v \in V}\). We show that all 4-edge-connected HHD-free graphs are \({\mathbb{Z}_3}\)-connected. This extends the result due to Lai (Graphs Comb 16:165–176, 2000), which proves the \({\mathbb{Z}_3}\)-connectivity for 4-edge-connected chordal graphs.

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  1. Graduate School of Informatics, Kyoto University, Kyoto, Japan

    Takuro Fukunaga

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  1. Takuro Fukunaga
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Correspondence to Takuro Fukunaga.

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Fukunaga, T. All 4-Edge-Connected HHD-Free Graphs are \({\mathbb{Z}_3}\)-Connected. Graphs and Combinatorics 27, 647–659 (2011). https://doi.org/10.1007/s00373-010-0995-9

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  • DOI: https://doi.org/10.1007/s00373-010-0995-9

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