In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional(k, m)-deleted and presents an...In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional(k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional(k, m)-deleted graphs improving the existing one. Finally, we state an open problem.展开更多
A spanning subgraph F of a graph G is called a path factor of G if each component of F is a path.A P≥k-factor means a path factor with each component having at least k vertices,where k≥2 is an integer.Bazgan,Benhamd...A spanning subgraph F of a graph G is called a path factor of G if each component of F is a path.A P≥k-factor means a path factor with each component having at least k vertices,where k≥2 is an integer.Bazgan,Benhamdine,Li and Wozniak[C.Bazgan,A.H.Benhamdine,H.Li,M.Wozniak,Partitioning vertices of 1-tough graph into paths,Theoret.Comput.Sci.263(2001)255–261.]obtained a toughness condition for a graph to have a P≥3-factor.We introduce the concept of a P≥k-factor deleted graph,that is,if a graph G has a P≥k-factor excluding e for every e∈E(G),then we say that G is a P≥k-factor deleted graph.In this paper,we show four sufficient conditions for a graph to be a P≥3-factor deleted graph.Furthermore,it is shown that four results are best possible in some sense.展开更多
A k-regular spanning subgraph of graph G is called a k-factor of G. Graph G is called a k-deleted graph if G-e has a k-factor for each edge e. A graph G=(X,Y) with bipartition (X,Y) is called a bipartite graph if ever...A k-regular spanning subgraph of graph G is called a k-factor of G. Graph G is called a k-deleted graph if G-e has a k-factor for each edge e. A graph G=(X,Y) with bipartition (X,Y) is called a bipartite graph if every edge of G has one endpoint in X and the other in Y.It is proved that a bipartite graph G=(X,Y) with X=Y is a k-deleted graph if and only if kS≤r 1+2r 2+...+k(r k+...+r Δ)-ε(S) for all SX. Using this result we give a sufficient neighborhood condition for a bipartite to be a k-deleted graph.展开更多
基金partially supported by MINECO(Grant No.MTM2014–51891–P and Fundación Séneca de la Región de Murcia 19219/PI/14)National Science Foundation of China(Grant No.11401519)
文摘In computer networks, toughness is an important parameter which is used to measure the vulnerability of the network. Zhou et al. obtains a toughness condition for a graph to be fractional(k, m)-deleted and presents an example to show the sharpness of the toughness bound. In this paper, we remark that the previous example does not work and inspired by this fact, we present a new toughness condition for fractional(k, m)-deleted graphs improving the existing one. Finally, we state an open problem.
基金supported by Six Talent Peaks Project in Jiangsu Province,China(Grant No.JY–022)。
文摘A spanning subgraph F of a graph G is called a path factor of G if each component of F is a path.A P≥k-factor means a path factor with each component having at least k vertices,where k≥2 is an integer.Bazgan,Benhamdine,Li and Wozniak[C.Bazgan,A.H.Benhamdine,H.Li,M.Wozniak,Partitioning vertices of 1-tough graph into paths,Theoret.Comput.Sci.263(2001)255–261.]obtained a toughness condition for a graph to have a P≥3-factor.We introduce the concept of a P≥k-factor deleted graph,that is,if a graph G has a P≥k-factor excluding e for every e∈E(G),then we say that G is a P≥k-factor deleted graph.In this paper,we show four sufficient conditions for a graph to be a P≥3-factor deleted graph.Furthermore,it is shown that four results are best possible in some sense.
文摘A k-regular spanning subgraph of graph G is called a k-factor of G. Graph G is called a k-deleted graph if G-e has a k-factor for each edge e. A graph G=(X,Y) with bipartition (X,Y) is called a bipartite graph if every edge of G has one endpoint in X and the other in Y.It is proved that a bipartite graph G=(X,Y) with X=Y is a k-deleted graph if and only if kS≤r 1+2r 2+...+k(r k+...+r Δ)-ε(S) for all SX. Using this result we give a sufficient neighborhood condition for a bipartite to be a k-deleted graph.
