It is said that a graph G is independent-set-deletable factor-critical （in short, ID-factor-critical）, if, for everyindependent-set I which has the same parity as ｜V（G）｜, G - I has a perfect matching. A graph G ...It is said that a graph G is independent-set-deletable factor-critical （in short, ID-factor-critical）, if, for everyindependent-set I which has the same parity as ｜V（G）｜, G - I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The κ-th power of G, denoted by G^κ, is the graph with vertex set V（G） in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G^3 and T（G） （the total graph of G） are ID-factor-critical, and G^4 （when ｜V（G）｜ is even） is strongly IM-extendable; if G is 2-connected, then D^2 is ID-factor-critical.展开更多
A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted τ c (G), is the minimum cardinality of a clique-transversal set in G. In th...A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted τ c (G), is the minimum cardinality of a clique-transversal set in G. In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound. Also, we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.展开更多
LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)?(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-g...LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)?(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively.展开更多
基金Project supported by NSFC(10371112)NSFHN (0411011200)SRF for ROCS,SEM
文摘It is said that a graph G is independent-set-deletable factor-critical （in short, ID-factor-critical）, if, for everyindependent-set I which has the same parity as ｜V（G）｜, G - I has a perfect matching. A graph G is strongly IM-extendable, if for every spanning supergraph H of G, every induced matching of H is included in a perfect matching of H. The κ-th power of G, denoted by G^κ, is the graph with vertex set V（G） in which two vertices are adjacent if and only if they have distance at most k in G. ID-factor-criticality and IM-extendability of power graphs are discussed in this article. The author shows that, if G is a connected graph, then G^3 and T（G） （the total graph of G） are ID-factor-critical, and G^4 （when ｜V（G）｜ is even） is strongly IM-extendable; if G is 2-connected, then D^2 is ID-factor-critical.
基金the National Nature Science Foundation of China (Grant Nos.10571117,60773078)the Hong Kong Polytechnic University (Grant No.G-YX69) Shuguang Plan of Shanghai Education Development Foundation (Grant No.06SG42)
文摘A clique-transversal set D of a graph G is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted τ c (G), is the minimum cardinality of a clique-transversal set in G. In this paper we present the bounds on the clique-transversal number for regular graphs and characterize the extremal graphs achieving the lower bound. Also, we give the sharp bounds on the clique-transversal number for claw-free cubic graphs and we characterize the extremal graphs achieving the lower bound.
文摘LetG be a graph with vertex setV (G) and edge setE (G), and letg andf be two integer-valued functions defined on V(G) such thatg(x)?(x) for every vertexx ofV(G). It was conjectured that ifG is an (mg +m - 1,mf -m+1)-graph andH a subgraph ofG withm edges, thenG has a (g,f)-factorization orthogonal toH. This conjecture is proved affirmatively.
