| The L(2,1)-labeling problem is a vertex-labeling analog of Hale’s channel assignment problem which seeks to minimize the range of frequencies used while at the same time ensur-ing that transmitters which are sufficiently close together are assigned transmission frequencies which differ by no less than a prescribed amount.An L(2,1)-labeling of a graph G is a function f from the vertex set of G to the set of nonnegative integers such that|f(x)-f(y)|≥2if d(x,y)=1, and|f(x)-f(y)|>1if d(x, y)=2, where x, y denotes vertex of G respectively, d(x, y) denotes the distance between the pair of vertices x,y. These labelings have been used to model the channel assignment problem. An L(2,1)-labeling of a graph G that uses label in the set{0.1,...,κ}(not necessarily all of them) is called a k-labeling. The minimum k so that G has a k-labeling is called the lambda number of G and is denoted by λ(G). A λ(G)-abeling is referred to simply as a λ-labeling.According to the contents, this paper is divided into four chapters:In the first chapters, it introduces the background, significance of paper and the present studying of L(2,1)-abeling, hole index, island, island sequences and some preparative theories.In the latter chapters, it is studied that path covering of several class particular2-sparse graph with cycles. At the same time, we proof that complement of these particular2-sparse graph with cycles are connected graphs that admits at least two different island sequences. |
