Sufficient Conditions for a Graph to Be L-Path-Coverable or L-Edge-Hamiltonian in Terms of the First Zagreb Index

The first Zagreb index $M_{1}(H)$ is equal to the sum of squares of the degrees of the vertices of the underlying graph $H$. Finding sufficient conditions for graphs to possess certainproperties is an important and meaningful problem. In this paper, we give sufficient conditions in terms of the first Zagreb index for a graph to be ${\ell}$-path-coverable or ${\ell}$-edge-Hamiltonian. All of these results are best possible.