Geodesic
Turán numbers and anti-Ramsey numbers for short cycles in complete \(3\)-partite graphs
The electronic journal of combinatorics, Tome 30 (2023) no. 2
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We call a $4$-cycle in $K_{n_{1}, n_{2}, n_{3}}$ multipartite (denoted by $C_{4}^{\text{multi}}$) if it contains at least one vertex in each part of $K_{n_{1}, n_{2}, n_{3}}$ . The Turán number of $ C_{4}^{\text{multi}})$ and $\{C_{3}, C_{4}^{\text{multi}}\})$ is determined in the paper as well as the anti-Ramsey number $\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})$. We prove that $\text{ex}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=n_{1}n_{2}+2n_{3}$ and$\text{ar}(K_{n_{1},n_{2},n_{3}}, C_{4}^{\text{multi}})=\text{ex}(K_{n_{1},n_{2},n_{3}}, \{C_{3}, C_{4}^{\text{multi}}\})+1=n_{1}n_{2}+n_{3}+1,$ where $n_{1}\ge n_{2}\ge n_{3}\ge 1.$
DOI : 10.37236/10628
Classification : 05C30, 05C15, 05C35, 05C38, 05C75
Mots-clés : multipartite cycles, anti-Ramsey number, Turán numbers 
@article{10_37236_10628,
     author = {Chunqiu Fang and Ervin Gy\H{o}ri and Chuanqi Xiao and Jimeng Xiao},
     title = {Tur\'an numbers and {anti-Ramsey} numbers for short cycles in complete \(3\)-partite graphs},
     journal = {The electronic journal of combinatorics},
     year = {2023},
     volume = {30},
     number = {2},
     doi = {10.37236/10628},
     zbl = {1516.05099},
     url = {http://geodesic.mathdoc.fr/articles/10.37236/10628/}
}
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Chunqiu Fang; Ervin Győri; Chuanqi Xiao; Jimeng Xiao. Turán numbers and anti-Ramsey numbers for short cycles in complete \(3\)-partite graphs. The electronic journal of combinatorics, Tome 30 (2023) no. 2. doi: 10.37236/10628

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