A star forest is a forest whose components are stars. The star arboricity of a graph G,denoted by sa( G),is the minimum number of star forests needed to decompose G. Let k be a positive integer. A k-star forest is a forest whose components are stars of order at most k + 1. The k-star arboricity of a graph G,denoted by sak( G),is the minimum number of k-star forests needed to decompose G. In this paper,it is proved that if any two vertices of degree 3 are nonadjacent in a subcubic graph G then sa2( G) ≤2.For general subcubic graphs G, a polynomial-time algorithm is described to decompose G into three 2-star forests. For a tree T andΔ( a positive integer k, T)it is proved that≤ sakk( T) ≤Δ( T)- 1+ 1,where Δ( T) is the maximum degree of T.kMoreover,a linear-time algorithm is designed to determine whether sak( T) ≤m for any tree T and any positive integers m and k.
For a graph G and two positive integers j and k, an m-L(j, k)-edge-labeling of G is an assignment on the edges to the set {0, 1, 2,..., m}, such that adjacent edges which receive labels differ at least by j, and edges which are distance two apart receive labels differ at least by kThe λ j,k-number of G is the minimum m such that an m-L(j, k)-edge-labeling is admitted by GIn this article, the L(1, 2)-edge-labeling for the hexagonal lattice, the square lattice and the triangular lattice are studied, and the bounds for λ j,k-numbers of these graphs are obtained.
