Linear Arboricity of the Tensor Products of Graphs

The linear arboricity, la(G), of a graph G is the minimum number of linear forests which partition the edge set of G. Akiyama et al. conjectured that la(G) = inverted right perpendicular Delta(G)+1/2 inverted left perpendicular for any regular graph G. In this paper, we prove this conjecture for K-m x K-n and K-m,K-m x K-n, where x denotes the tensor product of graphs. As a consequence, the above conjecture has been verified to be true for G x H, for any pair of graphs G and H, with Delta(G) = m - 1 and Delta(H) = n - 1, where m and n are the numbers of vertices of G and H, respectively.