The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. In this paper, it is proved that if G is a planar graph with ∆(G) ≥ 5 and without 4-cycles, then la(G) = ⌈∆(G) 2 ⌉. Moreover, the bound that ∆(G)≥ 5 is sharp.

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. In this paper, it is proved that for a planar graph G with maximum degree ∆(G) ≥ 7, la(G) = d 2 e if G has no 5-cycles with chords.

The linear arboricity la(G) of a graph G is the minimum number of linear forests which partition the edges of G. In this paper, it is proved that for a planar graph G with maximum degree ∆(G)≥ 7, la(G) = d(∆(G))/2e if G has no 5-cycles with chords. 2010 Mathematics Subject Classification: 05C15

We consider the problem of ranking assignments according to cost in the classical linear assignment problem. An algorithm partitioning the set of possible assignments, as suggested by Murty, is presented where, for each partition, the optimal assignment is calculated using a new reoptimization technique. Computational results for the new algorithm are presented.