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Coloring, sparseness and girth

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Abstract

An r-augmented tree is a rooted tree plus r edges added from each leaf to ancestors. For d, g, r ∈ N, we construct a bipartite r-augmented complete d-ary tree having girth at least g. The height of such trees must grow extremely rapidly in terms of the girth.

Using the resulting graphs, we construct sparse non-k-choosable bipartite graphs, showing that maximum average degree at most 2(k - 1) is a sharp sufficient condition for k-choosability in bipartite graphs, even when requiring large girth. We also give a new simple construction of non-k-colorable graphs and hypergraphs with any girth.

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Author information

Authors and Affiliations

  1. Sackler School of Mathematics and Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv, 69978, Israel

    Noga Alon

  2. School of Mathematics, Institute for Advanced Study, Princeton, NJ, 08540, USA

    Noga Alon

  3. Departments of Mathematics, University of Illinois, 1409 W. Green st., Urbana, IL, 61801, USA

    Alexandr Kostochka

  4. Zhejiang Normal University, China 688 Yingbin Road, Jinhua Zhejiang Province, China

    Alexandr Kostochka

  5. Department of Mathematics, University of Illinois, 1409 W. Green st., Urbana, IL, 61801, USA

    Benjamin Reiniger

  6. Departments of Mathematics, Zhejiang Normal University, China 688 Yingbin Road, Jinhua Zhejiang Province, China

    Douglas B. West

  7. University of Illinois, 1409 W. Green st., Urbana, IL, 61801, USA

    Douglas B. West

  8. Department of Mathematics, Zhejiang Normal University, China 688 Yingbin Road, Jinhua Zhejiang Province, China

    Xuding Zhu

Authors
  1. Noga Alon
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  2. Alexandr Kostochka
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  3. Benjamin Reiniger
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  4. Douglas B. West
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  5. Xuding Zhu
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Corresponding author

Correspondence to Noga Alon.

Additional information

Research supported in part by a USA-Israeli BSF grant, by an ISF grant, by the Israeli I-Core program and by the Oswald Veblen Fund.

Research supported in part by NSF grant DMS-1266016.

Research supported by Recruitment Program of Foreign Experts, 1000 Talent Plan, State Administration of Foreign Experts Affairs, China.

Research supported by CNSF 11571319.

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Alon, N., Kostochka, A., Reiniger, B. et al. Coloring, sparseness and girth. Isr. J. Math. 214, 315–331 (2016). https://doi.org/10.1007/s11856-016-1361-2

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  • DOI: https://doi.org/10.1007/s11856-016-1361-2

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