Gracefulness of two nested cycles: a first approach

Miguel Licona; Joaquín Tey

paper

Gracefulness of two nested cycles: a first approach

arXiv:2411.12998

Abstract

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 $M \equiv 0, 3 (mod 4)$ , 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 $m_{1} + m_{2} \equiv 0, 3 (mod 4)$ (resp. $m_{1} + m_{2} \equiv 1, 2 (mod 4)$ ), then there exists a graceful (resp. near-graceful) plane graph consisting of two nested cycles with sizes $m_{1}$ and $m_{2}$ , respectively.

24 pages, 10 figures