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On a Coloring Conjecture of Hajós

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Abstract

Hajós conjectured that graphs containing no subdivision of \(K_5\) are 4-colorable. It is shown in Yu and Zickfeld (J Comb Theory Ser B 96:482–492, 2006) that if there is a counterexample to this conjecture then any minimum such counterexample must be 4-connected. In this paper, we further show that if \(G\) is a minimum counterexample to Hajós’ conjecture and \(S\) is a 4-cut in \(G\) then \(G-S\) has exactly two components.

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Acknowledgments

We thank an anonymous referee for discovering a missing case in the original manuscript, and for an observation that led to a more direct argument in Case (4.1) in Sect. 3.

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Authors and Affiliations

  1. School of Mathematics and Physics, Shanghai University of Electric Power, Shanghai, 200090, China

    Yuqin Sun

  2. School of Mathematics, Georgia Institute of Technology, Atlanta, GA, 30332-0160, USA

    Xingxing Yu

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  1. Yuqin Sun
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  2. Xingxing Yu
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Correspondence to Xingxing Yu.

Additional information

Partially supported by NSF Grants DMS-1265564 and AST-1247545 and NSA Grant H98230-13-1-0255.

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Sun, Y., Yu, X. On a Coloring Conjecture of Hajós. Graphs and Combinatorics 32, 351–361 (2016). https://doi.org/10.1007/s00373-015-1539-0

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  • DOI: https://doi.org/10.1007/s00373-015-1539-0

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