The Research On Vertex-Graceful And Super Vertex-Graceful Labeling Of Graphs A(m,n)

So far, although the development history of the labeled graphs have been only for decades, but it is still difficult to study the labelings of general graphs in theory, researchers only discussed labelings of some special graphs, most of the literature have given labelings of a classes of graph.In practical applications, the researchers of labeled graphs proposed many labeling conc-ept according to practical problems. By these concepts, they changed these problems into mathematical models, then realized the purpose of solving issues by solving these mathemati-cal models.This article made a thorough research on a few class labelings of graph A(m,n). By the method of construction law and mathematics inference, this paper introduced the odd graceful labelings and vertex-graceful labelings of graph A(m,1) and A(m,n) in the certain conditions, researched on super vertex-graceful labelings of graph A(m,1), A(m,2) and A(m,n).My jobs are as follows:In this paper, we first introduced the results on gracefulness of graph A(m,n) under certain conditions. this paper gave the graceful labeling of graph A(m,1) when m=0(mod 4) and m=3(mod 4) by the method of construction law and mathematics inference, proved A(m,1) is odd graceful when m≥4 and m≡0(mod 2).The article reseached the vetex-graceful labelings of graph A(m,1) when m=1(mod 2) and the results are generalized to the vetex-gracefulness of graph A(m,1) when m= 1(mod 2) and n>1.The article demonstrated the conclution that A(m,1)is super vertex-graceful graph when m=0(mod 4)and m 2(mod 4)and proved graph A(m,2)is super vertex-graceful when m= 1(mod 2), researched on the super vertex-graceful labelings of graph A(m,n) when m=3,5, 7 and n=0(mod 2) in detail,gave the conclution that A(m,n) is super vertex-graceful when m=4 orm=6, for any positive integern.