Graphs with large maximum forcing number
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866912791633330176 |
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| author | Liu, Qianqian Diwan, Ajit A. Zhang, Heping |
| author_facet | Liu, Qianqian Diwan, Ajit A. Zhang, Heping |
| contents | For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq \frac{n^2}{n-F(G)}$, where $e(G)$ denotes the number of edges of $G$. In this paper we confirm this conjecture and obtain $F(G)\leq n-\frac{n^2}{e(G)}$. If $F(G)=n-1$, Liu and Zhang [9] proved that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along 4-cycles. If $G$ is bipartite and $F(G)\geq n-k$, $1\leq k\leq n-1$, we show that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along even cycles of length at most $2(k+1)$. Finally, we ask whether $f(G)\geq \lceil\frac{n}{k}\rceil-1$ holds for such bipartite graphs $G$, and give positive answers for the cases $k=1,2$. Further we show all minimum forcing numbers of the bipartite graphs $G$ of order $2n$ and with $F(G)=n-2$ form an integer interval $[\lfloor\frac{n}{2}\rfloor, n-2]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22761 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Graphs with large maximum forcing number Liu, Qianqian Diwan, Ajit A. Zhang, Heping Combinatorics For a graph $G$ with order $2n$ and a perfect matching, let $f(G)$ and $F(G)$ denote the minimum and maximum forcing number of $G$ respectively. Then $0\leq f(G)\leq F(G)\leq n-1$. Liu and Zhang [10] ever proposed a conjecture: $e(G)\geq \frac{n^2}{n-F(G)}$, where $e(G)$ denotes the number of edges of $G$. In this paper we confirm this conjecture and obtain $F(G)\leq n-\frac{n^2}{e(G)}$. If $F(G)=n-1$, Liu and Zhang [9] proved that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along 4-cycles. If $G$ is bipartite and $F(G)\geq n-k$, $1\leq k\leq n-1$, we show that any two perfect matchings of $G$ can be obtained from each other by a series of matching switches along even cycles of length at most $2(k+1)$. Finally, we ask whether $f(G)\geq \lceil\frac{n}{k}\rceil-1$ holds for such bipartite graphs $G$, and give positive answers for the cases $k=1,2$. Further we show all minimum forcing numbers of the bipartite graphs $G$ of order $2n$ and with $F(G)=n-2$ form an integer interval $[\lfloor\frac{n}{2}\rfloor, n-2]$. |
| title | Graphs with large maximum forcing number |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2512.22761 |