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A Variation of the Erdős–Sós Conjecture in Bipartite Graphs

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Abstract

The Erdős–Sós Conjecture states that every graph with average degree more than \(k-2\) contains all trees of order k as subgraphs. In this paper, we consider a variation of the above conjecture: studying the maximum size of an (n, m)-bipartite graph which does not contain all (k, l)-bipartite trees for given integers \(n\ge m\) and \(k\ge l\). In particular, we determine that the maximum size of an (n, m)-bipartite graph which does not contain all (n, m)-bipartite trees as subgraphs (or all (k, 2)-bipartite trees as subgraphs, respectively). Furthermore, all these extremal graphs are characterized.

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Acknowledgements

The authors would like to thank the anonymous referee for many helpful and constructive suggestions to an earlier version of this paper, in particular for giving new short proofs of Lemmas 2.6 and 2.7.

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Authors and Affiliations

  1. School of Mathematical Sciences, MOE-LSC, SHL-MAC, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai, 200240, People’s Republic of China

    Long-Tu Yuan & Xiao-Dong Zhang

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  1. Long-Tu Yuan
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  2. Xiao-Dong Zhang
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Correspondence to Xiao-Dong Zhang.

Additional information

This work is supported by the Joint NSFC-ISF Research Program (jointly funded by the National Natural Science Foundation of China and the Israel Science Foundation (No. 11561141001)), the National Natural Science Foundation of China (Nos. 11531001 and 11271256), Innovation Program of Shanghai Municipal Education Commission (No. 14ZZ016) and Specialized Research Fund for the Doctoral Program of Higher Education (No. 20130073110075).

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Yuan, LT., Zhang, XD. A Variation of the Erdős–Sós Conjecture in Bipartite Graphs. Graphs and Combinatorics 33, 503–526 (2017). https://doi.org/10.1007/s00373-017-1767-6

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  • DOI: https://doi.org/10.1007/s00373-017-1767-6

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