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Berge–Fulkerson Coloring for Infinite Families of Snarks

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Abstract

It is conjectured by Berge and Fulkerson that every bridgeless cubic graph has six perfect matchings such that each edge is contained in exactly two of them. Hägglund constructed Blowup\((K_4, C)\) and Blowup(Prism, \(C_4)\). Based on these two graphs, Chen constructed infinite families of bridgeless cubic graphs \(M_{0,1,2, \ldots ,k-2, k-1}\) which are obtained from cyclically 4-edge-connected and admitting Fulkerson-cover cubic graphs \(G_0,G_1,\ldots , G_{k-1}\) by recursive process. He obtained that every graph in \(M_{0,1,2,3}\) has a Fulkerson-cover and gave the open problem that whether every graph in \(M_{0,1,2, \ldots ,k-2, k-1}\) has a Fulkerson-cover. In this paper, we solve this problem and prove that every graph in \(M_{0,1,2, \ldots ,k-2, k-1}\) has a Fulkerson-cover.

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Acknowledgements

Many thanks to the anonymous referees for their helpful comments and suggestions. This work was supported by the NSFC (nos. 11371052,11171020), the Fundamental Research Funds for the Central Universities (nos. 2016JBM 071, 2016JBZ012), the 111 Project of China (B16002).

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Authors and Affiliations

  1. Department of Mathematics, Beijing Jiaotong University, Beijing, 100044, People’s Republic of China

    Ting Zheng & Rong-Xia Hao

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  1. Ting Zheng
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  2. Rong-Xia Hao
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Correspondence to Rong-Xia Hao.

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Communicated by Hossein Hajiabolhassan.

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Zheng, T., Hao, RX. Berge–Fulkerson Coloring for Infinite Families of Snarks. Bull. Iran. Math. Soc. 44, 277–290 (2018). https://doi.org/10.1007/s41980-018-0019-8

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