On path factor covered graphs with special constraints

Source abstract

For a family of connected graphs $\mathcal{H}$, an $\mathcal{H}$-factor is a spanning subgraph of a graph, whose connected components are isomorphic to graphs from the set $\mathcal{H}$. A graph $G$ is called an $\mathcal{H}$-factor covered graph if there is an $\mathcal{H}$-factor of $G$ covering $e$ for any $e\in E(G)$. Furthermore, a graph $G$ is called an $(\mathcal{H},l)$-factor deleted covered graph if for every $E'\subseteq E(G)$ with $|E'|=l, G-E'$ is an $\mathcal{H}$-factor covered graph. The structure of a graph is usually measured by graph parameters such as toughness, isolated toughness, binding number and degree sum, etc. In this paper, we use these graph parameters to study $(\mathscr{P}_{\geq 2},l)$-factor deleted covered graphs and $(\mathscr{P}_{\geq 3},l)$-factor deleted covered graphs, respectively.