Gracefulness of two nested cycles: a first approach

TL;DR

This paper investigates the gracefulness of plane graphs with two nested cycles, establishing conditions under which their semiduals are conservative or near-conservative, and demonstrating existence results for such graphs based on cycle sizes.

Contribution

It provides a first approach to understanding the gracefulness of nested cycle graphs and characterizes the semiduals' properties based on cycle sizes modulo 4.

Findings

Semidual of nested cycle graphs is conservative if total size ≡ 0,3 mod 4.

Existence of graceful or near-graceful nested cycle graphs depends on cycle sizes and their sum modulo 4.

Conditions for the existence of such graphs are established for large cycle sizes.

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 $M \equiv 0,3 \pmod 4$, 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.