Asian Research Journal of Mathematics

Let G be a simple, finite, connected, undirected, non-trivial graph with  p vertices and q edges. V(G) be the vertex set and E(G) be the edge set of G. The n^th Hilbert number is denoted by H_n and is defined by H_n = 4(n-1)+1 where n \(\ge\) 1. A Hilbert  graceful  labeling is an injective function H from the vertex set V(G) to a set of Hilbert number {x : x = 4(i-1)+1,1 \(\le\) i \(\le\) 2q} which induces a bijective function H* from the set E(G) to the set of number {1,2,3,4,......,q }, where for each edge uv \(\in\) E(G) with u,v \(\in\) V(G) applies H*(uv)= \(\frac{1}{4}\)|H(u) - H(v)|. A graph with Hilbert graceful labeling is called a Hilbert graceful graph. This research aims to prove that some complete multipartite graphs are Hilbert graceful by providing systematic proofs and clear constructions of the labeling functions. Our contributions include the identification and characterization of these graphs, expanding the class of graphs known to exhibit Hilbert graceful properties, and providing illustrative examples to support our findings.