Extension of a list coloring problem

For a graph H , f (H ) is the smallest integer k such that the join of H with an empty graph Ek of order k is not |V (H )|-choosable. It was conjectured that for a triangle-free graph G, f (G) = ( 2 )μ(G) nn−2μ(G), where n = |V (G)| and μ(G) is the cardinality of a maximum matching of graph G [S. Gravier, F. Maffray, B. Mohar, On a list-coloring problem, Discrete Math. 268 (2003) 303–308]. We verify this conjecture in the case of forests, and propose some related problems. © 2005 Published by Elsevier Ltd