The Minimum Degree And Order Of The Strongly Z_2p+1-Connected Graphs

Let G be an undirected graph,then G is strongly Z2p+1-connected if for every map b:V(G)→Z2p+1 with ∑v∈V(G)b(v)≡0(mod 2p+1),there is an orientation D of G satisfying dD+(v)-dD-(v)≡b(v)(mod 2p+1)for every v∈E V(G).This paper proves some theorems about strongly Z2p+1-connectedness and considered how the minimum degree and order of a graph and its complementary graph infect these theorems.The paper consists of the following three parts:The first chapter introduces a short survey of Theorems about group connectivity and the concept of connectivity,Zk-flow,Zk-connectivity,nowhere zero Zk-flow,strongly Zk-connectivity and unitary Zk-flow(or mod k-orientation).As the these concepts are,an analysis is studied in 1.3,Some notations and symbols are also given.In the second chapter,firstly,it narrates the theorems about the strongly Z3-connected graphs;Secondly,we prove that if a simple graph has a nontrivial strongly Z3-connected subgraph and min{δ(G),δ(Gc)}≥4,then either G or Gc is strongly Z3-connected,this result is useful for proving later theorems;thirdly,the result that if a simple graph G with|v(G)|≥44 and min{δ(G),δ(Gc)}≥4,then either G or Gc is strongly Z3-connected is proved in[3],and we improve the result that if a simple graph G with |v(G)|≥18 and min{δ(G),δ(Gc)}≥ 4,then either G or Gc is strongly Z3-connected,it reduces the order from 44 to 18;fourthly,we consider the simple bipartite graph that if |V(G)|≥18 and min{δ(G),δ(Gc)}≥ 3,then either G or Gc is strongly Z3-connected.In the third chapter,firstly,it narrates the theorems about the strongly Z2p+1-connected graphs;Secondly,we prove that K4p+1,4p+2∈M2p+10,and on the basis of this result,we get other strongly Z2p+1-connected bipartite graphs;thirdly,the result that if a simple graph G with |V(G)|≥1152p4 and min{δ(G),δ(Gc)}≥4p,then either G or Gc is strongly Z3-connected is proved in[5],and we improve the result that if a simple graph G with|V(G)|≥8p2+26p+1 and min{δ(G),δ(Gc)}≥ 4p,then either G or Gc is strongly Z2p+1-connected,it reduces the order from 1152p4 to 8p2+26p+1;fourthly,we consider the simple bipartite graph that if |V(G)|≥12p and min{δ(G),δ(Gc)} 4p,then ei-ther G or Gc is strongly Z2p+i-connected;fifthly,we consider that if p=2,as long as min{δ(G),δ(Gc)}≥8,then either G or Gc is strongly Z5-connected;lastly,whether the results about minimum degree and order are optimum are given.