Let G=(V,E)be a graph.For a vertex labeling f:V→Z2,it induces an edge labeling f+:E→Z2,where for each edge v1 v2∈E we have f+(v1 v2)=f(v1)+f(v2).For each i∈Z2,we use vf(i)(respectively,ef(i))to denote the number o...Let G=(V,E)be a graph.For a vertex labeling f:V→Z2,it induces an edge labeling f+:E→Z2,where for each edge v1 v2∈E we have f+(v1 v2)=f(v1)+f(v2).For each i∈Z2,we use vf(i)(respectively,ef(i))to denote the number of vertices(respectively,edges)with label i.A vertex labeling f of G is said to be friendly if vertices with different labels differ in size by at most one.The full friendly index set of a graph G,denoted by F F I(G),consists of all possible values of ef(1)-ef(0),where f ranges over all friendly labelings of G.In this paper,motivated by a problem raised by[6],we study the full friendly index sets of a family of cubic graphs.展开更多
Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i...Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.展开更多
Let G be a connected simple graph with vertex set V(G)and edge set E(G).A binary vertex labeling f:V(G)→Z2,is said to be friendly if the number of vertices with different labels differs by at most one.Each vertex fri...Let G be a connected simple graph with vertex set V(G)and edge set E(G).A binary vertex labeling f:V(G)→Z2,is said to be friendly if the number of vertices with different labels differs by at most one.Each vertex friendly labeling/induces an edge labeling f*E(G)→Z2,defined by f*(xy)=f(x)+f(y)for each xy∈E(G).Let er(i)=\{e∈E(G):f*(e)=i}|.The full friendly index set of G,denoted by FFI(G),is the set{ef*(1)-ep(0):f is friendly}.In this paper,we determine the full friendly index set of a family of cycle union graphs which are edge subdivisions of P2×Pn.展开更多
In this paper, we introduce the concept of the general butterfly graph B[m,n;d] for integers m,n ≥ 3, d ≥ 1, determine its balance index set, and give the necessary and sufficient condition for balanced graph B[m,n;...In this paper, we introduce the concept of the general butterfly graph B[m,n;d] for integers m,n ≥ 3, d ≥ 1, determine its balance index set, and give the necessary and sufficient condition for balanced graph B[m,n;d] to exist.展开更多
基金Supported by the National Natural Science Foundation of China(Grant No.11801149)Doctoral Fund of Henan Polytechnic University(Grant No.B2018-55)。
文摘Let G=(V,E)be a graph.For a vertex labeling f:V→Z2,it induces an edge labeling f+:E→Z2,where for each edge v1 v2∈E we have f+(v1 v2)=f(v1)+f(v2).For each i∈Z2,we use vf(i)(respectively,ef(i))to denote the number of vertices(respectively,edges)with label i.A vertex labeling f of G is said to be friendly if vertices with different labels differ in size by at most one.The full friendly index set of a graph G,denoted by F F I(G),consists of all possible values of ef(1)-ef(0),where f ranges over all friendly labelings of G.In this paper,motivated by a problem raised by[6],we study the full friendly index sets of a family of cubic graphs.
基金Supported by FRG/07-08/II-08 Hong Kong Baptist University
文摘Let G =（V, E） be a connected simple graph. A labeling f ： V → Z2 induces an edge labeling f＊ ： E → Z2 defined by f＊（xy） = f（x） ＋f（y） for each xy ∈ E. For i ∈ Z2, let vf（i） = ｜f^-1（i）｜ and ef（i） = ｜f＊^-1（i）｜. A labeling f is called friendly if ｜vf（1） - vf（0）｜ ≤ 1. For a friendly labeling f of a graph G, we define the friendly index of G under f by if（G） = e（1） - el（0）. The set [if（G） ｜ f is a friendly labeling of G} is called the full friendly index set of G, denoted by FFI（G）. In this paper, we will determine the full friendly index set of every Cartesian product of two cycles.
基金This work was supported partly by the National Natural Science Foundation of China(Grant Nos.11801149,11801148)S.Wu was also partially supported by the Doctoral Fund of Henan Polytechnic University(B2018-55).
文摘Let G be a connected simple graph with vertex set V(G)and edge set E(G).A binary vertex labeling f:V(G)→Z2,is said to be friendly if the number of vertices with different labels differs by at most one.Each vertex friendly labeling/induces an edge labeling f*E(G)→Z2,defined by f*(xy)=f(x)+f(y)for each xy∈E(G).Let er(i)=\{e∈E(G):f*(e)=i}|.The full friendly index set of G,denoted by FFI(G),is the set{ef*(1)-ep(0):f is friendly}.In this paper,we determine the full friendly index set of a family of cycle union graphs which are edge subdivisions of P2×Pn.
基金the National Natural Science Foundation of China （No. 10671005） the Natural Science Foundation of Hebei Province （No. A2007000230）.
文摘In this paper, we introduce the concept of the general butterfly graph B[m,n;d] for integers m,n ≥ 3, d ≥ 1, determine its balance index set, and give the necessary and sufficient condition for balanced graph B[m,n;d] to exist.
