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The Conjecture on the Crossing Number of \(K_{1,m,n}\) is true if Zarankiewicz’s Conjecture Holds

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Abstract

Zarankiewicz’s conjecture states that the crossing number \(\text {cr}(K_{m,n})\) of the complete bipartite graph \(K_{m,n}\) is \(Z(m,n):=\lfloor \frac{m}{2}\rfloor \lfloor \frac{m-1}{2}\rfloor \lfloor \frac{n}{2}\rfloor \lfloor \frac{n-1}{2}\rfloor\), where \(\lfloor x \rfloor\) denotes the largest integer no more than x. It is conjectured that the crossing number \(\text {cr}(K_{1,m,n})\) of the complete tripartite graph \(K_{1,m,n}\) is \(Z(m+1,n+1)-\lfloor \frac{m}{2}\rfloor \lfloor \frac{n}{2}\rfloor\). When one of m and n is even, Ho proved that this conjecture is true if Zarankiewicz’s conjecture holds, in 2008. When both m and n are odd, Ho proved that \(\text {cr}(K_{1,m,n})\ge \text {cr}(K_{m+1,n+1})-\left\lfloor \frac{n}{m}\lfloor \frac{m}{2}\rfloor \lfloor \frac{m+1}{2}\rfloor \right\rfloor\) and conjectured that equality holds in this inequality. Which one of the conjectures may be true? In this paper, we proved that if Zarankiewicz’s conjecture holds, then the former one is true.

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Acknowledgements

The authors are indebted to two anonymous referees for their thoughtful comments and suggestions, which improved the presentation, made it more readable and shortened the manuscript a great deal. The work was supported by the National Natural Science Foundation of China (No. 61401186).

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Correspondence to Xiwu Yang.

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Yang, X., Wang, Y. The Conjecture on the Crossing Number of \(K_{1,m,n}\) is true if Zarankiewicz’s Conjecture Holds. Graphs and Combinatorics 37, 1083–1088 (2021). https://doi.org/10.1007/s00373-021-02303-y

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