The existence of a path-factor without small odd paths
Summary: A \(\{P_{2},P_{5}\}\)-factor of a graph is a spanning subgraph of the graph each of whose components is isomorphic to either \(P_{2}\) or \(P_{5}\), where \(P_{n}\) denote the path of order \(n\).~In this paper, we show that if a graph \(G\) satisfies \(c_{1}(G-X)+\frac{2}{3}c_{3}(G-X)\leq \frac{4}{3}|X|+\frac{1}{3}\) for all \(X\subseteq V(G)\), then \(G\) has a \(\{P_{2},P_{5}\}\)-factor, where \(c_{i}(G-X)\) is the number of components \(C\) of \(G-X\) with \(|V(C)|=i\). Moreover, it is shown that above condition is sharp.
scientific article; zbMATH DE number 6873084