The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs
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arXiv
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2025
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| _version_ | 1866908483958341632 |
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| author | Yu, Yuantian Li, Shuchao |
| author_facet | Yu, Yuantian Li, Shuchao |
| contents | Given a graph $H,$ we say that a graph is \textit{$H$-free} if it does not contain $H$ as a subgraph. The Turán number $\ex(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding extremal graphs is denoted by $\Ex(n, H)$. The study of Turán number of graphs is a central topic in extremal graph theory. A graph is \textit{color-critical} if it contains an edge whose deletion reduces its chromatic number. Simonovits showed that if $H$ is a color-critical graph of chromatic number $r+1,$ then for sufficiently large $n,$ $\Ex(n, H)=\{T_r(n)\},$ the $r$-partite Turán graph of order $n.$ Given a color-critical graph $H$ with chromatic number $r+1,$ it is interesting to determine $H$-free non-$r$-partite graphs with maximum number of edges. For a graph $H$ with chromatic number $r+1,$ denote $\ex_{r+1}(n,H)$ the maximum number of edges in non-$r$-partite $H$-free graphs of order $n,$ the set of all non-$r$-partite $H$-free graphs of order $n$ and size $\ex_{r+1}(n,H)$ is denoted by $\Ex_{r+1}(n, H)$. For $r\geq 3,\,k\geq1,$ the generalized book graph \({B}_{r,k}\) is a graph obtained by joining every vertex of $K_r$ to every vertex of an independent set of size \(k\). Note that \({B}_{r,k}\) is a color-critical graph of chromatic number $r+1.$ In this paper, based on the stability theory and local structure characterization, the exact value of $\ex_{r+1}(n,B_{r,k})$ is determined and all the corresponding extremal graphs are identified, where $r\geq 3,\,k\geq1$ and $n$ is sufficiently large. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_07533 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs Yu, Yuantian Li, Shuchao Combinatorics 05C50, 05C75 Given a graph $H,$ we say that a graph is \textit{$H$-free} if it does not contain $H$ as a subgraph. The Turán number $\ex(n,H)$ of $H$ is the maximum number of edges in an $n$-vertex $H$-free graph, the set of all the corresponding extremal graphs is denoted by $\Ex(n, H)$. The study of Turán number of graphs is a central topic in extremal graph theory. A graph is \textit{color-critical} if it contains an edge whose deletion reduces its chromatic number. Simonovits showed that if $H$ is a color-critical graph of chromatic number $r+1,$ then for sufficiently large $n,$ $\Ex(n, H)=\{T_r(n)\},$ the $r$-partite Turán graph of order $n.$ Given a color-critical graph $H$ with chromatic number $r+1,$ it is interesting to determine $H$-free non-$r$-partite graphs with maximum number of edges. For a graph $H$ with chromatic number $r+1,$ denote $\ex_{r+1}(n,H)$ the maximum number of edges in non-$r$-partite $H$-free graphs of order $n,$ the set of all non-$r$-partite $H$-free graphs of order $n$ and size $\ex_{r+1}(n,H)$ is denoted by $\Ex_{r+1}(n, H)$. For $r\geq 3,\,k\geq1,$ the generalized book graph \({B}_{r,k}\) is a graph obtained by joining every vertex of $K_r$ to every vertex of an independent set of size \(k\). Note that \({B}_{r,k}\) is a color-critical graph of chromatic number $r+1.$ In this paper, based on the stability theory and local structure characterization, the exact value of $\ex_{r+1}(n,B_{r,k})$ is determined and all the corresponding extremal graphs are identified, where $r\geq 3,\,k\geq1$ and $n$ is sufficiently large. |
| title | The exact Turán number of generalized book graph $B_{r,k}$ in non-$r$-partite graphs |
| topic | Combinatorics 05C50, 05C75 |
| url | https://arxiv.org/abs/2508.07533 |