Given a graph G, a subgraph C is called a clique of G if C is a complete subgraph of G maximal under inclusion and ｜C｜ ≥2. A clique-transversal set S of G is a set of vertices of G such that S meets all cliques of ...Given a graph G, a subgraph C is called a clique of G if C is a complete subgraph of G maximal under inclusion and ｜C｜ ≥2. A clique-transversal set S of G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted as τC（G）, is the minimum cardinality of a clique-transversal set in G. The clique-graph of G, denoted as K（G）, is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques in G have nonempty intersection. Let F be a class of graphs G such that F = {G｜ K（G） is a tree}. In this paper the graphs in F having independent clique-transversal sets are shown and thus τC（G）/｜G｜ ≤ 1/2 for all G ∈F.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
A clique-transversal set D of a graph C is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by To（G）, is the minimum cardinality of a clique- transversal set in G. In...A clique-transversal set D of a graph C is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by To（G）, is the minimum cardinality of a clique- transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.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 by τC（G）,is the minimum cardinality of a clique-transversal set in G.In thi...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 by τC（G）,is the minimum cardinality of a clique-transversal set in G.In this paper,we first present a lower bound on τC（G） and characterize the extremal graphs achieving the lower bound for a connected（claw,K4）-free 4-regular graph G.Furthermore,we show that for any 2-connected（claw,K4）-free 4-regular graph G of order n,its clique-transversal number equals to [n/3].展开更多
基金Project supported by the National Natural Science Foundation of China （Grant No.10571117）, and the Development Foundation of Shanghai Municipal Commission of Education （Grant No.05AZ04）
文摘Given a graph G, a subgraph C is called a clique of G if C is a complete subgraph of G maximal under inclusion and ｜C｜ ≥2. A clique-transversal set S of G is a set of vertices of G such that S meets all cliques of G. The clique-transversal number, denoted as τC（G）, is the minimum cardinality of a clique-transversal set in G. The clique-graph of G, denoted as K（G）, is the graph obtained by taking the cliques of G as vertices, and two vertices are adjacent if and only if the corresponding cliques in G have nonempty intersection. Let F be a class of graphs G such that F = {G｜ K（G） is a tree}. In this paper the graphs in F having independent clique-transversal sets are shown and thus τC（G）/｜G｜ ≤ 1/2 for all G ∈F.
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.
基金Supported by National Natural Science Foundation of China （Grant No. 60773078）, the PuJiang Project of Shanghai （Grant No. 09PJ1405000） and Key Disciplines of Shanghai Municipality （Grant No. $30104）
文摘A clique-transversal set D of a graph C is a set of vertices of G such that D meets all cliques of G. The clique-transversal number, denoted by To（G）, is the minimum cardinality of a clique- transversal set in G. In this paper we give the exact value of the clique-transversal number for the line graph of a complete graph. Also, we give a lower bound on the clique-transversal number for 4-regular claw-free graphs and characterize the extremal graphs achieving the lower bound.
基金Supported by National Nature Science Foundation of China(Grant Nos.11171207 and 10971131)
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 by τC（G）,is the minimum cardinality of a clique-transversal set in G.In this paper,we first present a lower bound on τC（G） and characterize the extremal graphs achieving the lower bound for a connected（claw,K4）-free 4-regular graph G.Furthermore,we show that for any 2-connected（claw,K4）-free 4-regular graph G of order n,its clique-transversal number equals to [n/3].
