Some Results on Felicitous Labeling of Graphs

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 \( mn \equiv 2 \pmod{4} \).
2. \( mC_n \) is felicitous if and only if \( mn \not\equiv 2 \pmod{4} \).

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 \( n \), \( mC_n \) is felicitous if \( mn \equiv 1, 3 \pmod{4} \).
4. For any positive integer \( m \), \( mC_4 \) is felicitous.