Edge Fault Tolerance For Regular Graphs Of Optimally H-restricted Edge-connectivity

The connectivity of graph G=(V(G),E(G))is an important index of network reliability.However,for modern networks,the classical connectivity can no longer satisfy the need we want to study.Therefore,we refer the concepts of restricted edge-connectivity of Fabrega and Fiol(undirected,and restricted arc-connectivity of Zhang et al.for digraphs)and fault tolerance of Zhao et al.An edge subset F(?)E(G)is an h-restricted edge cut if G-F is disconnected and every component of G-F has at least h vertices.However,such an h-restricted edge cut does not always exist,the h-restricted edge-connectivity of G,if any,denoted by ?^(h)(G),is defined to be the cardinality of a minimum h-restricted edge cut.Let?h(G)=min{|E[X,X]|:X(?)V(G),|X|=h and G[X]is connected}.Zhang et al.give a sufficient condition which guarantees the inequality ?^(h)(G)??h(G).In this case,a graph G is ?^(h)?optimal if ?^(h)(G)=?h(G).The edge fault tolerance ?^(h)(G)of a ?^(h)-optimal graph G is the maximum integer m for which G-F is still ?^(h)-optimal for any F(?)E(G)with |F|?m.This paper mainly determines the bounds of fault tolerance of the undirected regular graph for ?^(h)-optimal.For digraph,this paper gives the restricted arc-connectivity of Harary digraphs.