Purwanto, Rossi Dwi Iswanto
In this paper all graphs are finite, simple, and nonempty. Let G be a graph having vertex set V(G), edge set E(G), and number of vertices q. A graceful labeling f of G is an injective function f:V(G)→{0,1,2,..,q} such that the induced edge labeling f*:E(G)→{1,2,..,q}, defined by f*(xy)=|f(x)-f(y)| is bijective. A graph that can be labeled by a graceful labeling is said to be graceful. In 1972, Golomb proved that the complete graph Km is graceful if and only if m≤4. For m≥3, A lollipop graph Lm,n is a graph obtained from a complete graph Km and a path Pn by joining one vertex of Km to one end vertex of Pn by an edge. A tied lollipop graph TLm,n,t is a graph obtained from lollipop graph Lm,n and t edges by identifying one vertex of each of the edge with the vertex of degree m in Lm,n. In this paper we study graceful labeling of tied graceful graphs and find that some of these graphs are graceful. © 2023 Author(s).
Department of Mathematics, Universitas Negeri Malang, Jalan Semarang 5, East Java, Malang, 65145, Indonesia
