Low chromatic spanning sub(di)graphs with prescribed degree or connectivity properties

Abstract Generalizing well‐known results of Erdős and Lovász, we show that every graph contains a spanning ‐partite subgraph with , where is the edge‐connectivity . In particular, together result due to Nash‐Williams Tutte, this implies 7‐edge‐connected bipartite whose edge set decomposes into two edge‐disjoint trees. We best possible as it does not hold for infinitely many 6‐edge‐connected graphs. For directed graphs, was shown by Bang‐Jensen et al. there no such ‐arc‐connected digraph has strong subdigraph. prove 3‐partite subdigraph semicomplete on at least six vertices generalize higher connectivities proving that, positive integer ()‐partite which possible. A conjecture Kreutzer minimum out‐degree bound would be providing an infinite class digraphs do contain any in all out‐degrees are also semidegree 6‐partite vertex in‐