Gracefulness of \(n\) – Cone \(C_m \vee K_n^c\)

Abstract

It is proved that the \(n\)-cone \(C_m \vee K_n^c\) is graceful for any \(n \geq 1\) and \(m = 0\) or \(3 \pmod{12}\). The gracefulness of the following \(n\)-cones is also established: \(C_4 \vee K_n^c\), \(C_5 \vee K_2^c\), \(C_7 \vee K_n^c\), \(C_9 \vee K_2^c\), \(C_{11} \vee K_n^c\), \(C_{19} \vee K_n^c\). This partially answers the question of gracefulness of \(n\)-cones which is listed as an open problem in the survey article by J.A. Gallian.