On The Graceful Of Union Graph Of Paths, Circles

The graceful graph is one of an important role in the research subjects of graph theory. There are wide ranges of appliations not only in the graph graph itself but also in other fields, such as radio astronomy, x-radial crystalloid study, radar, circuit design, communication networks. Therefore, the graceful graph is obtained professors'attention quickly as the branch of graph theory. In this paper, we main discuss the graceful labelings of union graph of paths and circles.For G = (V ,E), If for each one v∈V, there exists a non-negative integerθ(v) (called the vertices v of the label), meet: (1) (?)u , v∈V, if u≠v, thenθ(u )≠θ(v); (2) max{θ(v )| v∈V}=|E|; (3) (?)e₁ , e₂∈E, if e₁≠e₂, thenθ′( e₁ )≠θ′(e₂), whichθ′( e )=|θ(u )-θ(v)|, e= uv. G is called graceful graph.θis called graceful value or graceful labeling,θ′known as the edge of G induced value byθ.In this paper, we discuss three kinds of graceful union graphs, such as the union of path and path, circle and circle, path and circle. The main research is that structure graceful labelings of some union graph, we pay our attention mainly on the following three domination.First, we discuss the gracefulness on union of paths and paths. Take the structure method, graceful labelings of P_{k , 2,n} and P_{k ,m,n} are got. and we proved that are all graceful.Second, we discuss the gracefulness on union of circles and circles. Take the structure method, graceful labelings of C_{6 ,i,2n}(i =1,2,3), C_{8 ,i,n}(i =1,2,3,4),C_{12 ,i,n}(i =1,2,…,6) , C_{16 ,i,n}(i =1,2,…,8)C_4k,i,n(i =2k ,2k-1,2k-2) are got. and we proved that are all graceful. And guess that C_4k,i,n are graceful graphs.Third, we discuss the gracefulness on union of paths and circles. Take the structure method, graceful labelings of (C_n + C_m)·P_l+1, and are got. and we proved that are all graceful.