Abstract
Letp≧2 be a fixed integer, and letG be a connected graph onn vertices. Ifδ(G)≧2, ifd(u)+d(v)>2n/p−2 holds wheneveruv∉E(G), and ifn is sufficiently large compared top, then eitherG has a spanning eulerian subgraph, orG is contractible to a graphG 1 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.
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References
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Catlin, P.A. Contractions of graphs with no spanning eulerian subgraphs. Combinatorica 8, 313–321 (1988). https://doi.org/10.1007/BF02189088
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DOI: https://doi.org/10.1007/BF02189088