A Note on the Omitted Vertex Label of Graceful Pendant Graphs

A graceful labeling of a graph \( G \) with \( q \) edges is an injective assignment from the vertices of \( G \) into \(\{0, 1, \ldots, q\}\) such that when each edge is assigned the absolute value of the difference of the vertex labels it connects, the resulting edge labels are distinct. In 1978, Frucht conjectured that for gracefully labeled coronas \( C_n \odot K_1 \), the omitted vertex label is always even. In this paper, we will verify Frucht’s conjecture.