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Turán Numbers of Several Bipartite Graphs

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Abstract

For graphs \(H_1,H_2,\dots ,H_k\), the k-color Turán number \(ex(n,H_1,H_2,\dots ,H_k)\) is the maximum number of edges in a k-colored graph of order n that does not contain monochromatic \(H_i\) in color i as a subgraph, where \(1\le i\le k\). In this note, we show that if \(H_i\) is a bipartite graph with at least two edges for \(1\le i\le k\), then \(ex(n,H_1,H_2,\dots ,H_k)=(1+o(1))\sum _{i=1}^kex(n,H_i)\) as \(n\rightarrow \infty \), in which the non-constructive proof for some cases can be derandomized.

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Authors and Affiliations

  1. College of Mathematical Sciences, Harbin Engineering University, Harbin, 150001, China

    Ye Wang

  2. School of Mathematical Sciences, Tongji University, Shanghai, 200092, China

    Yusheng Li

  3. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Yan Li

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  1. Ye Wang
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  2. Yusheng Li
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  3. Yan Li
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Correspondence to Yan Li.

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Supported by Natural Science Foundation of Heilongjiang Province of China (LH2021A004). Supported by NSFC (11871377,11931002). Supported by NSFC (12301451).

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Wang, Y., Li, Y. & Li, Y. Turán Numbers of Several Bipartite Graphs. Graphs and Combinatorics 40, 3 (2024). https://doi.org/10.1007/s00373-023-02731-y

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