The complexity of finding low chromatic spanning sub(di)graphs with prescribed connectivity properties

As usual λ(G) denotes the edge-connectivity of graph G. It was shown in [2] that every G contains a spanning (λ(G)+1)-partite subgraph H such λ(H)=λ(G) and one can find polynomial time. We determine complexity deciding, for given positive integers r,k whether r-colourable which is k-edge-connected. show problem polynomially solvable when r>k NP-complete otherwise. In fact, combined with result from above, this means precisely r k-edge-connected has One all graphs whose edge set decomposes into k edge-disjoint trees are 2k-colourable. consider deciding collection union forms an r≥2k other values r. also analogous digraph D arc-disjoint out-branchings subdigraph formed by arcs branchings r-colourable.