Gracefulness of two nested cycles: a first approach
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916488715173888 |
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| author | Licona, Miguel Tey, Joaquín |
| author_facet | Licona, Miguel Tey, Joaquín |
| contents | It is known that if a plane graph is graceful (resp. near-graceful), then its semidual is conservative (resp. near-conservative). In this work we prove that the semidual of a plane graph of size $M$ consisting of two nested cycles is conservative if $M \equiv 0,3 \pmod 4$, and near-conservative otherwise. We also show that for a given integer $m_1 \geq 3$, there exists $m^* > m_1$ such that for $m_2 \geq m^*$, if $m_1+m_2 \equiv 0,3 \pmod 4$ (resp. $m_1+m_2 \equiv 1,2 \pmod 4$), then there exists a graceful (resp. near-graceful) plane graph consisting of two nested cycles with sizes $m_1$ and $m_2$, respectively. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_12998 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gracefulness of two nested cycles: a first approach Licona, Miguel Tey, Joaquín Combinatorics 05C78, 05C05, 05C21 It is known that if a plane graph is graceful (resp. near-graceful), then its semidual is conservative (resp. near-conservative). In this work we prove that the semidual of a plane graph of size $M$ consisting of two nested cycles is conservative if $M \equiv 0,3 \pmod 4$, and near-conservative otherwise. We also show that for a given integer $m_1 \geq 3$, there exists $m^* > m_1$ such that for $m_2 \geq m^*$, if $m_1+m_2 \equiv 0,3 \pmod 4$ (resp. $m_1+m_2 \equiv 1,2 \pmod 4$), then there exists a graceful (resp. near-graceful) plane graph consisting of two nested cycles with sizes $m_1$ and $m_2$, respectively. |
| title | Gracefulness of two nested cycles: a first approach |
| topic | Combinatorics 05C78, 05C05, 05C21 |
| url | https://arxiv.org/abs/2411.12998 |