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On connencted $$[g,f+1]$$-factors in graphs. (English) Zbl 1090.05058
Let $$G=(V(G), E(G))$$ be a graph and $$g$$ and $$f$$ positive integral functions on $$V(G)$$ such that $$2 \leq g(v) \leq f(v) \leq d_G(v)$$ for all $$v$$. The spanning subgraph $$F$$ is a $$[g, f]$$-factor of $$G$$ if $$g(v) \leq d_F(v) \leq f(v)$$ for all $$v$$. The main result of the paper is that if $$G$$ has both a $$[g, f]$$-factor and a Hamiltonian path, then $$G$$ has a connected $$[g, f+1]$$-factor. It is a generalization of Kano’s conjecture first solved by M. Cai [Syst. Sci. Math. Sci. 8, No. 4, 364–368 (1995; Zbl 0851.05083)].

##### MSC:
 05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
##### Keywords:
$$[g, f]$$-factors; Hamiltonian graphs; Kano’s conjecture
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