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
Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P...Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P^((3))_(1)∨K_((2))^(2n))were presented.For natural number n∈N,n≥1,the two graphs are all graceful graphs,where P^((1))_(2n),P^((2))_(2n)are even-vertices path,P_(2n+1)is odd-vertices path,K_(2n),K^((1))_(2n),K^((2))_(2n)are the complement of graph K_(2 n),G_(1)∨G_(2)is the join graph of G_(1)and G_(2).展开更多
In the paper, we study the gracefulness of several unconnected graphs related to wheel. For natural number p ≥ 1, t ≥ 1 , let n = 2t ＋ 3,2t ＋ 4 , which proved W. U K （1） p,t U K（2） is graceful; for p≥1, t≥1 ...In the paper, we study the gracefulness of several unconnected graphs related to wheel. For natural number p ≥ 1, t ≥ 1 , let n = 2t ＋ 3,2t ＋ 4 , which proved W. U K （1） p,t U K（2） is graceful; for p≥1, t≥1 ,let n=2t＋3,2t＋4, then Wn,2n＋1 U K（1）p,t U K（2） p,t is graceful and for m ≥ 1, r ≥ 1 , let n = 2m ＋ 5, Wn,2n＋1 U （C3 v Km） U St（r） is graceful.展开更多
The present paper shows the coordinates of a tree and its vertic es, defines a kind of Trees with Odd-Number Radiant Type (TONRT), deals with th e gracefulness of TONRT by using the edge-moving theorem, and uses gra...The present paper shows the coordinates of a tree and its vertic es, defines a kind of Trees with Odd-Number Radiant Type (TONRT), deals with th e gracefulness of TONRT by using the edge-moving theorem, and uses graceful TON RT to construct another class of graceful trees.展开更多
LOCATED in GuizhouProvince, southwesternChina, Qianxinan Bouyei-Miao AutonomousPrefecture is a tourist destination,due to its extraordinary naturalenvironment and distinctive
The traditional gardens of Suzhou,Jiangsu Province,draw on Eastern concepts of harmonious living.Their roots reach back to the Spring and Autumn Period(770-476B.C.),when the city was founded.They flourished during the...The traditional gardens of Suzhou,Jiangsu Province,draw on Eastern concepts of harmonious living.Their roots reach back to the Spring and Autumn Period(770-476B.C.),when the city was founded.They flourished during the Ming and Qing dynasties(1368-1911),when Suzhou ranked among China's most prosperous cities.展开更多
Let G(V,E) be a simple graph and G^k be a k-power graph defined byV(G~*) = V(G), E(G^k) = E(G) ∪{uv|d(u,v) =k} for natural number k. In this paper,it is proved that P_n^3 is a graceful graph.
A digraph D（V, E） is said to be graceful if there exists an injection f ： V（G） →{0, 1,... , ｜E｜} such that the induced function f＇ ： E（G） --~ {1, 2,… , ｜E｜} which is defined by f＇ （u, v） = [f（v） - ...A digraph D（V, E） is said to be graceful if there exists an injection f ： V（G） →{0, 1,... , ｜E｜} such that the induced function f＇ ： E（G） --~ {1, 2,… , ｜E｜} which is defined by f＇ （u, v） = [f（v） - f（u）] （rood ｜E｜＋ 1） for every directed edge （u, v） is a bijection. Here, f is called a graceful labeling （graceful numbering） of D（V, E）, while f＇ is called the induced edge＇s graceful labeling of D. In this paper we discuss the gracefulness of the digraph n- Cm and prove that n. Cm is a graceful digraph for m = 15, 17 and even展开更多
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.
基金the National Natural Science Foundation of China(11702094)the Fundamental Research Funds for the Central University(3142015045)。
文摘Two kinds of unconnected double fan graphs with even vertices,(P^((1))_(1)∨(P^((1))_(2n)∪P^((2))_(2n)))∪P_(2n+1)∪(P_(1)^((2))∨K_(2n))and(P_(1)^((1))∨(P^((1))_(2n)∪P^((2))_(2n)))∪(P_(1)^((2))∨K_((1))^(2n))∪(P^((3))_(1)∨K_((2))^(2n))were presented.For natural number n∈N,n≥1,the two graphs are all graceful graphs,where P^((1))_(2n),P^((2))_(2n)are even-vertices path,P_(2n+1)is odd-vertices path,K_(2n),K^((1))_(2n),K^((2))_(2n)are the complement of graph K_(2 n),G_(1)∨G_(2)is the join graph of G_(1)and G_(2).
基金Supported by the Natural Science Foundation of Beijing(1102015)University Scientific Research Project of Hebei Province(Z2014032)the Fundamental Research Funds for the Central Universities(HKXJZD201402,2011B019,3142013025,3142014127)
文摘In the paper, we study the gracefulness of several unconnected graphs related to wheel. For natural number p ≥ 1, t ≥ 1 , let n = 2t ＋ 3,2t ＋ 4 , which proved W. U K （1） p,t U K（2） is graceful; for p≥1, t≥1 ,let n=2t＋3,2t＋4, then Wn,2n＋1 U K（1）p,t U K（2） p,t is graceful and for m ≥ 1, r ≥ 1 , let n = 2m ＋ 5, Wn,2n＋1 U （C3 v Km） U St（r） is graceful.
文摘The present paper shows the coordinates of a tree and its vertic es, defines a kind of Trees with Odd-Number Radiant Type (TONRT), deals with th e gracefulness of TONRT by using the edge-moving theorem, and uses graceful TON RT to construct another class of graceful trees.
文摘LOCATED in GuizhouProvince, southwesternChina, Qianxinan Bouyei-Miao AutonomousPrefecture is a tourist destination,due to its extraordinary naturalenvironment and distinctive
文摘The traditional gardens of Suzhou,Jiangsu Province,draw on Eastern concepts of harmonious living.Their roots reach back to the Spring and Autumn Period(770-476B.C.),when the city was founded.They flourished during the Ming and Qing dynasties(1368-1911),when Suzhou ranked among China's most prosperous cities.
文摘Let G(V,E) be a simple graph and G^k be a k-power graph defined byV(G~*) = V(G), E(G^k) = E(G) ∪{uv|d(u,v) =k} for natural number k. In this paper,it is proved that P_n^3 is a graceful graph.
文摘A digraph D（V, E） is said to be graceful if there exists an injection f ： V（G） →{0, 1,... , ｜E｜} such that the induced function f＇ ： E（G） --~ {1, 2,… , ｜E｜} which is defined by f＇ （u, v） = [f（v） - f（u）] （rood ｜E｜＋ 1） for every directed edge （u, v） is a bijection. Here, f is called a graceful labeling （graceful numbering） of D（V, E）, while f＇ is called the induced edge＇s graceful labeling of D. In this paper we discuss the gracefulness of the digraph n- Cm and prove that n. Cm is a graceful digraph for m = 15, 17 and even
