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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2301.13632 |
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| _version_ | 1866912364848218112 |
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| author | Boyer, Geoffrey Goddard, Wayne |
| author_facet | Boyer, Geoffrey Goddard, Wayne |
| contents | We contribute results on $r$-regular graphs that do and don't have the maximum possible toughness, namely $r/2$. Doty and Ferland showed the existence of a $5$-regular graph with toughness $5/2$ for all even orders except $n= 18$. Using a computer search we show that there does not exist such a graph for $n=18$. Also, we provide the first family of $4$-regular graphs with toughness $2$ that contains claws. For the prism $G \Box K_2$ of a graph~$G$, we provide several bounds including a sufficient condition for the prism to have the same toughness as~$G$. In particular, we show that if $G$ has toughness $t\le \frac{1}{2}$ then its prism has toughness $2t$; further, the prism of any $r$-regular $r$-connected inflation has toughness~$r/2$ (despite being $(r+1)$-regular) and in general the prism of any $3$-regular graph has toughness at most~$3/2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2301_13632 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | On the Toughness of Regular Graphs and Prisms Boyer, Geoffrey Goddard, Wayne Combinatorics 05C42 We contribute results on $r$-regular graphs that do and don't have the maximum possible toughness, namely $r/2$. Doty and Ferland showed the existence of a $5$-regular graph with toughness $5/2$ for all even orders except $n= 18$. Using a computer search we show that there does not exist such a graph for $n=18$. Also, we provide the first family of $4$-regular graphs with toughness $2$ that contains claws. For the prism $G \Box K_2$ of a graph~$G$, we provide several bounds including a sufficient condition for the prism to have the same toughness as~$G$. In particular, we show that if $G$ has toughness $t\le \frac{1}{2}$ then its prism has toughness $2t$; further, the prism of any $r$-regular $r$-connected inflation has toughness~$r/2$ (despite being $(r+1)$-regular) and in general the prism of any $3$-regular graph has toughness at most~$3/2$. |
| title | On the Toughness of Regular Graphs and Prisms |
| topic | Combinatorics 05C42 |
| url | https://arxiv.org/abs/2301.13632 |