Tight bounds for the existence of path factors in network vulnerability parameter settings

The issues of ruggedness and vulnerability are cruxes in network security research, which must be considered during the network designing phase. Parameters such as toughness, isolated toughness, and binding number characterize the vulnerable of the network from the structure of networks. The path factor, a special case of the generalized Script capital H-factor, measures the feasibility of data transmission in networks. Recent advances have been obtained to show that there is an inevitable connection between the vulnerability parameters of the network and the existence of path factors, while we found that some existing theoretical results are not tight and there is still a long way for further improvement. In view of graph theory approaches, this paper mainly contributes to determine the sharp bounds of toughness, isolated toughness, and binding number for the existence of path factor in different settings, and therefore solve the open problems left unsolved in previous articles.