Gracefulness of two nested cycles: A first approach

Abstract

It is known that if a plane graph admits a graceful (resp. near-graceful) labeling, then its semidual admits a conservative (resp. near-conservative) labeling. Consequently, we studied gracefulness of two nested cycles graphs considering two different perspectives: the first one by finding graceful (near-graceful) labelings of two nested cycles graphs, and the other one by finding conservative (near-conservative) labelings of its semidual. In this work we prove that for a given integer $m_1\ge 3$, there exists an integer $m^{\ast }>m_1$ such that for all $m_2\ge m^{\ast }$, if $m_1+m_2\equiv 0\text{or}3(mod4)$ (resp. $m_1+m_2\equiv 1,2(mod4)$), then there exists a graceful (resp. near-graceful) plane cycle with chords consisting of two nested cycles of sizes $m_1$ and $m_2$, respectively. We also show that the semidual of a plane cycle with chords of size $M$ consisting of two nested cycles is conservative if $M\equiv 0\text{or}3(mod4)$, and it is near-conservative otherwise.