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. A 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 d_G(u,v)=1 and |f(u)-f(v)|≥1 if d_G(u,v)=2. The L(2,1)-labeling numberλ(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 numbers and L(2,1)-labeling numbers of graphs. Our main work extends most of the results in [S.S. Adams, A. Trazkovich, D.S. Troxell, B. Westgate, On island sequences of labelings with a condition at distance two, Discrete Appl. Math. 158 (2010) 1-7] and can answer an open problem in [J. Georges, D.W. Mauro, On the structure of graphs with non-surjective L(2, 1)-labelings, SIAM J. Discrete Math. 19 (2005) 208-223].