Factors and vertex-deleted subgraphs (Q2488936)

Let \(G\) be a graph of order at least \(2\) and let \(f\) be an integer-valued function defined on the vertex set \(V(G)\) of \(G\) such that \(\sum_{x\in V(G)}f(x)\) is even. \textit{H. Enomoto} and \textit{T. Tokuda} [Discrete Math. 220, 239--242 (2000; Zbl 0955.05079)] proved that if \(G-x\) has an \(f\)-factor, for each \(x\in V(G)\), then \(G\) has an \(f\)-factor. The following theorem generalizing this result is the main theorem of the paper. Let \(G\) be a graph without isolated vertices, let \(X\) be a subset of \(V(G)\) such that \(| V(G)-X| \geq 2\) and let \(f\) be an integer-valued function defined on \(V(G)\) such that \(\sum_{x\in V(G)}f(x)\) is even. Moreover, let \(\sum_{x\in X}\deg_G(x)\leq 2| V(G)-X| +| X| -3\), when \(| X| \not= 1\) and \(\sum_{x\in X}\deg_G(x)\leq 2| V(G)-X| -1\), when \(| X| = 1\). If \(G-x\) has an \(f\)-factor, for each \(x\in V(G)-X\), then \(G\) has an \(f\)-factor.