On the Strongly \(c\)-Harmoniousness of Cycles with Some Chords

Abstract

For \(1 \leq s \leq n-3\), let \(C_n(i;i_1, v_2, \ldots, i_s)\) denote an \(n\)-cycle with consecutive vertices \(x_1, x_2, \ldots, x_n\) to which the \(s\) chords \(x_{ i}x_{i_1}, x_{i}x_{i_2}, \ldots, x_{i}x_{i_s}\) have been added. In this paper, we discuss the strongly \(c\)-harmonious problem of the graph \(C_n(i;i_1, i_2, \ldots, i_s)\).

A shell of width \(n\) is a fan \(C_n(1;3,4, \ldots, n-1)\) and a vertex with degree \(n-1\) is called apex. \(MS(n^m)\) is a graph consisting of \(m\) copies of shell of width \(n\) having a common apex. If \(m \geq 1\) is odd, then the multiple shell \(MS(n^ m)\) is harmonious.