Neighborhood condition for all fractional (g, f, n′, m)-critical deleted graphs

Abstract In data transmission networks, the availability of data transmission is equivalent to the existence of the fractional factor of the corresponding graph which is generated by the network. Research on the existence of fractional factors under specific network structures can help scientists design and construct networks with high data transmission rates. A graph G is named as an all fractional (g, f, n′, m)-critical deleted graph if the remaining subgraph keeps being an all fractional (g, f, m)-critical graph, despite experiencing the removal of arbitrary n′ vertices of G. In this paper, we study the relationship between neighborhood conditions and a graph to be all fractional (g, f, n′, m)-critical deleted. Two sufficient neighborhood conditions are determined, and furthermore we show that the conditions stated in the main results are sharp.