The minimum size and maximum diameter of an edge-pancyclic graph of a given order
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909349870305280 |
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| author | Li, Chengli Liu, Feng Zhan, Xingzhi |
| author_facet | Li, Chengli Liu, Feng Zhan, Xingzhi |
| contents | A $k$-cycle in a graph is a cycle of length $k.$ A graph $G$ of order $n$ is called edge-pancyclic if for every integer $k$ with $3\le k\le n,$ every edge of $G$ lies in a $k$-cycle. It seems difficult to determine the minimum size $f(n)$ of a simple edge-pancyclic graph of order $n.$ We give lower and upper bounds on $f(n),$ and determine the maximum diameter of such a graph. In the $3$-connected case, the precise value of $f(n)$ is determined. We also determine the minimum size of a graph of a given order with connectivity conditions in which every edge lies in a triangle. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11183 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The minimum size and maximum diameter of an edge-pancyclic graph of a given order Li, Chengli Liu, Feng Zhan, Xingzhi Combinatorics 05C35, 05C38, 05C45 A $k$-cycle in a graph is a cycle of length $k.$ A graph $G$ of order $n$ is called edge-pancyclic if for every integer $k$ with $3\le k\le n,$ every edge of $G$ lies in a $k$-cycle. It seems difficult to determine the minimum size $f(n)$ of a simple edge-pancyclic graph of order $n.$ We give lower and upper bounds on $f(n),$ and determine the maximum diameter of such a graph. In the $3$-connected case, the precise value of $f(n)$ is determined. We also determine the minimum size of a graph of a given order with connectivity conditions in which every edge lies in a triangle. |
| title | The minimum size and maximum diameter of an edge-pancyclic graph of a given order |
| topic | Combinatorics 05C35, 05C38, 05C45 |
| url | https://arxiv.org/abs/2410.11183 |