List total arboricity of 2-degenerate graphs

The vertex arboricity ρ ( G ) of a graph G is the smallest number of colours required to colour the vertices of G such that no cycle is monochromatic. The list vertex arboricity ρ l ( G ) is the list-colouring version of this concept. In this paper it is proved for the total graph T ( G ) of G that if G is a 2 -degenerate graph with maximum degree Δ ( G ) , then ⌈ ( Δ ( G ) + 1 ) / 2 ⌉ ≤ ρ ( T ( G ) ) ≤ ρ l ( T ( G ) ) ≤ ⌈ ( Δ ( G ) + 2 ) / 2 ⌉ . This shows that ρ ( T ( G ) ) = ρ l ( T ( G ) ) when Δ ( G ) is even. ve further that ρ ( T ( G ) ) = ρ l ( T ( G ) ) = ⌈ ( Δ ( G ) + 1 ) / 2 ⌉ if G is a cycle, or a tree with Δ ( G ) ≥ 2 .