Linear arboricity for graphs with multiple edges

Akiyama, Exoo, and Harary conjectured that for any simple graph G with maximum degree Δ( G ), the linear arboricity la(G) satisfies ⌈Δ( G )/2⌉ ≦ la(G) ≦ ⌈(Δ( G ) + 1)/2⌉. Here it is proved that if G is a loopless graph with maximum degree Δ( G ) ≦ k and maximum edge multiplicity μ( G) ≦ k − 2 n +1 + 1, where k ≧ 2 n −2 , then la(G) ≦ k − 2 n . It is also conjectured that for any loopless graph G , ⌈Δ( G )/2⌉ ≦ la(G) ≦ ⌈(Δ( G ) + μ( G ))/2⌉.