Some Results on Felicitous Labeling of Graphs

Abstract

Figueroa-Centeno, Ichishima, and Muntaner-Batle [3, 4] proved some results on felicitous graphs and raised the following conjectures:

1. The one-point union of $m$ copies of $C_n$ is felicitous if and only if $mn\equiv 2(\mathrm{mod}4)$.
2. $m{C}_n$ is felicitous if and only if $mn\not\equiv 2(\mathrm{mod}4)$.

In this paper, the conjectures are partially settled by proving the following results:

1. For any odd positive integers $m$ and $n$, the one-point union of $m$ copies of $C_n$ is felicitous if $mn\equiv 1,3$.
2. For any positive integer $m$, the one-point union of $m$ copies of $C_4$ is felicitous.
3. For any two odd positive integers $m$ and $n$, $m{C}_n$ is felicitous if $mn\equiv 1,3(\mathrm{mod}4)$.
4. For any positive integer $m$, $m{C}_4$ is felicitous.