Contractions of graphs with no spanning eulerian subgraphs

Letp2 be a fixed integer, and letG be a connected graph onn vertices. If(G)2, ifd(u)+d(v)>2n/p–2 holds wheneveruvE(G), and ifn is sufficiently large compared top, then eitherG has a spanning eulerian subgraph, orG is contractible to a graphG₁ of order less thenp and with no spanning eulerian subgraph. The casep=2 was proved by Lesniak-Foster and Williamson. The casep=5 was conjectured by Benhocine, Clark, Köhler, and Veldman, when they proved virtually the casep=3. The inequality is best-possible.