Path covering number and L(2,1)-labeling number of graphs

A path covering of a graph G is a set of vertex disjoint paths of G containing all the vertices of G. The path covering number of G, denoted by P (G), is the minimum number of paths in a path covering of G. An k-L(2, 1)-labeling of a graph G is a mapping f from V (G) to the set {0, 1, . . . , k} such that |f(u) − f(v)| ≥ 2 if dG(u, v) = 1 and |f(u) − f(v)| ≥ 1 if dG(u, v) = 2. The L(2, 1)-labeling number λ(G) of G is the smallest number k such that G has a k-L(2, 1)-labeling. The purpose of this paper is to study path covering number and L(2, 1)-labeling number of graphs. Our main work extends most of results in [On island sequences of labelings with a condition at distance two, Discrete Applied Maths 158 (2010), 1-7] and can answer an open problem in [On the structure of graphs with non-surjective L(2, 1)-labelings, SIAM J. Discrete Math. 19 (2005), 208-223].