Ore‐type condition for the existence of connected factors

In this article, we obtain some Ore‐type sufficient conditions for a graph to have a connected factor with degree restrictions. Let α and k be positive integers with $\alpha \ge {{k + 1} \over{k - 1}}$ if ${{k}} \ge 2$ and $\alpha \ge 4$ if ${{k}}=1$ . Let G be a connected graph with a spanning subgraph F , each component of which has order at least α. We show that if the degree sum of two nonadjacent vertices is greater than ${{2(\vert {G} \vert-\alpha-1)} \over {k+1}}$ then G has a connected subgraph in which F is contained and every vertex $v$ has degree at most ${\rm deg}_{{F}}(v)\,+\,{{k}}$ . From the result, we derive that a graph G has a connected $[{{a}},{{b}}]$ ‐factor $({{b}}>{{a}})$ if the degree sum of two nonadjacent vertices is at least ${{2[(a+1)\vert{G}\vert-(a+2)]} \over {a + b}}$ . © Wiley Periodicals, Inc. J. Graph Theory 56: 241–248, 2007