On the strongly $c$-harmoniousness cycle with $P_2-$ or $P_3$-chord

Abstract

The graph $C_n(d;i,j;P_k)$ denotes a cycle $C_n$ with path $P_k$ joining two nonconsecutive vertices $x_i$ and $x_j$ of the cycle, where $d$ is the distance between $x_i$ and $x_j$ on $C_n$. In this paper, we obtain that the graph $C_n(d;i,j;P_k)$ is strongly $c$-harmonious when $k=2,3$ and integer $n\ge 6$.