Diab proved the following graphs are Cordial;Pm K1,n if and only if(m,n) =(1,2);Cm K1,n;Pm Kn;Cm Kn for all m and n except m ≡ 2(mod 4).In this paper,we proved the Cordiality on the union of 3-regular connected graph...Diab proved the following graphs are Cordial;Pm K1,n if and only if(m,n) =(1,2);Cm K1,n;Pm Kn;Cm Kn for all m and n except m ≡ 2(mod 4).In this paper,we proved the Cordiality on the union of 3-regular connected graph K3 and cycle Cm.First we have the Lemma 2,if uv ∈ E(G),G is Cordial,we add 4 vertices x,y,z,w in sequence to the edge uv,obtain a new graph denoted by G*,then G* is still Cordial,by this lemma,we consider four cases on the union of 3-regular connected graph R3,and for every case we distinguish four subcases on the cycle Cm.展开更多
This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of ...This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appears in the graph. The paper examines the existence of such labelings for disjoint union of plane graphs.展开更多
Let a,b,k,r be nonnegative integers with 1 ≤ a ≤b and r ≥ 2. Let G be a graph of order n with n 〉 （a＋b）（r（a＋b）-2）＋ak/a. In this paper, we first show a characterization for all fractional （a, b, k）-criti...Let a,b,k,r be nonnegative integers with 1 ≤ a ≤b and r ≥ 2. Let G be a graph of order n with n 〉 （a＋b）（r（a＋b）-2）＋ak/a. In this paper, we first show a characterization for all fractional （a, b, k）-critical graphs. Then using the result, we prove that G is all fractional （a, b, k）-critical if δ（G） ≥ （r-1）b2/a ＋k and ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn＋ak/a＋b for any independent subset {xl, x2, .., xr} in G. Furthermore, it is shown that the lower bound on the condition ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn=ak/ a＋b is best possible in some sense, and it is an extension of Lu＇s previous result.展开更多
Combining forbidden subgraphs with degree restrictions and neighborhood unionrestrictions,respectively,we prove the following results:（1） Let G be a 2-connected graph of order n,and 3≤c≤n.If for each induced subgr...Combining forbidden subgraphs with degree restrictions and neighborhood unionrestrictions,respectively,we prove the following results:（1） Let G be a 2-connected graph of order n,and 3≤c≤n.If for each induced subgraphL of order four of G（?）|V<sub>1</sub>（L）∩S<sub>c</sub>|≥2 if L≌K<sub>1,3</sub>,and |V（L）∩S<sub>c</sub>|≥1 if L≌P<sub>4</sub>,then thecircumference of G is at least c,where V<sub>1</sub>（L）is the set of vertices with degree 1 of L,S<sub>c</sub> isthe set of vertices with degree at least c/2 of G and P<sub>4</sub> is a path of order 4.（2） Let G be a 2-connected graph of order n,and n≥s+2.If for each induced subgraphL of G isomorphic to K<sub>1,3</sub>or P<sub>4</sub>,d<sub>L</sub>（u,v）=2（?）|N（u）∪N（v）|≥s,then the circumferencec （G） of G is at least s+2.Moreover,if n≥s+3 and s is odd,then c（G）≥s+3.展开更多
The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a gracef...The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a graceful graph for m=j-1 or m≥n＋j,where C_（4n） is a cycle with 4n vertexes,P_m is a path with m＋1 vertexes,and（jC_（4n））∪P_m denotes the disjoint union of j-C_（4n） and P_m.展开更多
Let be an injective function. For a vertex labeling f, the induced edge labeling is defined by, or;then, the edge labels are distinct and are from . Then f is called a root square mean labeling of G. In this paper, we...Let be an injective function. For a vertex labeling f, the induced edge labeling is defined by, or;then, the edge labels are distinct and are from . Then f is called a root square mean labeling of G. In this paper, we prove root square mean labeling of some degree splitting graphs.展开更多
将文献[5](Shee S C,Ho YS.The Cordiality of the Path-union ofnCopies of a Graph.Discrete Math,1996,151:221-229.)的结果推广到Tn-union的情形,且不要求每个节点的图形必须相同.并给出了任意圈和扇Tn-union的Cordial性的分析和证明.
Let C be a 2-connected graph on > 2 31 venices. G is called pancyclic if itcontains a cycle of length I for every I such that 3 l n. In this paper we shall prove thatif IN(u) U N(v) Z (2n - 3)/3 for any nonadjacent...Let C be a 2-connected graph on > 2 31 venices. G is called pancyclic if itcontains a cycle of length I for every I such that 3 l n. In this paper we shall prove thatif IN(u) U N(v) Z (2n - 3)/3 for any nonadjacent pair uv E V(G), then G is pancyclic.展开更多
文摘Diab proved the following graphs are Cordial;Pm K1,n if and only if(m,n) =(1,2);Cm K1,n;Pm Kn;Cm Kn for all m and n except m ≡ 2(mod 4).In this paper,we proved the Cordiality on the union of 3-regular connected graph K3 and cycle Cm.First we have the Lemma 2,if uv ∈ E(G),G is Cordial,we add 4 vertices x,y,z,w in sequence to the edge uv,obtain a new graph denoted by G*,then G* is still Cordial,by this lemma,we consider four cases on the union of 3-regular connected graph R3,and for every case we distinguish four subcases on the cycle Cm.
文摘This paper deals with the problem of labeling the vertices, edges and faces of a plane graph in such a way that the label of a face and the labels of the vertices and edges surrounding that face add up to a weight of that face, and the weights of all s-sided faces constitute an arithmetic progression of difference d, for each s that appears in the graph. The paper examines the existence of such labelings for disjoint union of plane graphs.
基金Supported by National Natural Science Foundation of China(Grant No.11371009)
文摘Let a,b,k,r be nonnegative integers with 1 ≤ a ≤b and r ≥ 2. Let G be a graph of order n with n 〉 （a＋b）（r（a＋b）-2）＋ak/a. In this paper, we first show a characterization for all fractional （a, b, k）-critical graphs. Then using the result, we prove that G is all fractional （a, b, k）-critical if δ（G） ≥ （r-1）b2/a ＋k and ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn＋ak/a＋b for any independent subset {xl, x2, .., xr} in G. Furthermore, it is shown that the lower bound on the condition ｜NG（xl） ∪NG（x2） ∪... ∪NG（xr）｜ ≥ bn=ak/ a＋b is best possible in some sense, and it is an extension of Lu＇s previous result.
基金A work supported by National Natural Science Foundation of China
文摘Combining forbidden subgraphs with degree restrictions and neighborhood unionrestrictions,respectively,we prove the following results:（1） Let G be a 2-connected graph of order n,and 3≤c≤n.If for each induced subgraphL of order four of G（?）|V<sub>1</sub>（L）∩S<sub>c</sub>|≥2 if L≌K<sub>1,3</sub>,and |V（L）∩S<sub>c</sub>|≥1 if L≌P<sub>4</sub>,then thecircumference of G is at least c,where V<sub>1</sub>（L）is the set of vertices with degree 1 of L,S<sub>c</sub> isthe set of vertices with degree at least c/2 of G and P<sub>4</sub> is a path of order 4.（2） Let G be a 2-connected graph of order n,and n≥s+2.If for each induced subgraphL of G isomorphic to K<sub>1,3</sub>or P<sub>4</sub>,d<sub>L</sub>（u,v）=2（?）|N（u）∪N（v）|≥s,then the circumferencec （G） of G is at least s+2.Moreover,if n≥s+3 and s is odd,then c（G）≥s+3.
文摘The present paper deals with the gracefulness of unconnected graph （jC_（4n））∪P_m,and proves the following result：for positive integers n,j and m with n≥1,j≥2,the unconnected graph（jC_（4n））∪P_m is a graceful graph for m=j-1 or m≥n＋j,where C_（4n） is a cycle with 4n vertexes,P_m is a path with m＋1 vertexes,and（jC_（4n））∪P_m denotes the disjoint union of j-C_（4n） and P_m.
文摘Let be an injective function. For a vertex labeling f, the induced edge labeling is defined by, or;then, the edge labels are distinct and are from . Then f is called a root square mean labeling of G. In this paper, we prove root square mean labeling of some degree splitting graphs.
文摘将文献[5](Shee S C,Ho YS.The Cordiality of the Path-union ofnCopies of a Graph.Discrete Math,1996,151:221-229.)的结果推广到Tn-union的情形,且不要求每个节点的图形必须相同.并给出了任意圈和扇Tn-union的Cordial性的分析和证明.
文摘Let C be a 2-connected graph on > 2 31 venices. G is called pancyclic if itcontains a cycle of length I for every I such that 3 l n. In this paper we shall prove thatif IN(u) U N(v) Z (2n - 3)/3 for any nonadjacent pair uv E V(G), then G is pancyclic.
