Let G = (V (G),E(G)) be a graph with vertex set V (G) and edge set E(G), and g and f two positive integral functions from V (G) to Z +-{1} such that g(v) ≤ f(v) ≤ d G (v) for all v ∈V (G), where d G (v) is the degree of the vertex v. It is shown that every graph G, including both a [g,f]-factor and a hamiltonian path, contains a connected [g,f +1]-factor. This result also extends Kano’s conjecture concerning the existence of connected [k,k+1]-factors in graphs.
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* The work of this author was supported by NSFC of China under Grant No. 10271065, No. 60373025.
† The work of these authors was also supported in part by the US Department of Energy’s Genomes to Life program (http://doegenomestolife.org/) under project, “Carbon Sequestration in Synechococcus sp.: From Molecular Machines to Hierarchical Modeling” (www.genomes2life.org) and by National Science Foundation (NSF/DBI-0354771,NSF/ITR-IIS-0407204).
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Li*†, G., Xu†, Y., Chen, C. et al. On Connected [g,f +1]-Factors in Graphs. Combinatorica 25, 393–405 (2005). https://doi.org/10.1007/s00493-005-0023-9
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DOI: https://doi.org/10.1007/s00493-005-0023-9