Connected \([k,k+1]\)-factors of graphs
The author studies degree conditions for the existence of almost \(k\)-regular factors and connected \([k,k+1]\)-factors in graphs of order \(n\). He treats especially the case \(kn\) odd, continuing his previous work from [Syst. Sci. Math. Sci. 8, No. 4, 364-368 (1995; Zbl 0851.05083)]. He proves that if \(k\geq 3\) is odd, \(\varepsilon \in \{-1,+1\}\), and \(G\) is a connected graph of odd order \(n\geq 4k-3\) with minimum degree at least \(k\) and such that of any two nonadjacent vertices, at least one has degree larger than \({n\over 2}\), then \(G\) has a factor in which all vertices with one exception have degree \(k\), and the exceptional vertex has degree \(k+ \varepsilon\). The exceptional vertex can be chosen in advance from the vertices with degree larger than \({n \over 2}\). He also shows that by adding a properly chosen matching one can obtained a connected \([k,k+1]\)-factor, thereby proving a conjecture of Kano for \(kn\) odd.
scientific article; zbMATH DE number 1021668