Gracefully Harmonious Graphs

Labelling problems are very useful in modelling, solving problems in otherareas and enhance the study of algebraic structures. Nowadays, a more practical,accurate and efficient way of labeling a graph is using some mathematical methodslike integer programming or constraint programming. This will improve the scope andquality of research. Here, considering the applications of Graceful and Harmoniouslabelings, we introduce a few new types of labeling, namely, gracefully harmoniouslabeling, gracefully felicitous labeling, gracefully elegant labeling, gracefully stronglyc-elegant labeling and gracefully semi-harmonious labeling. In this paper, we provethat some complete multi-partite graphs is gracefully harmonious, gracefully felicitousand gracefully semi-harmonious.