Nowhere-Zero3-Flows And Group Connectivity In Graphs

Let G be a graph and D be an orientation of G. We denote ED/+(Ï…)(ED/+(Ï…)) by the edges with tail(head) at v, respectively. If there is a function f:E(G)â†'{±1,±2,...,±(k-1)} such that for every vertex υ∈V(G), then G admits a nowhere-zero k-flow.Let A be an abelian group with identity0. If for any function b:V(G)â†'A with Συ€V(G) b(v)=0> there is a function f:E(G)â†'Aâ†'{O} such that for each vertex υ€V(G),, then G is A-connected.The concept of nowhere-zero flow was first introduced by Tutte as a tool to attack the4-color-conjecture. In1976, he put forward the famous3-flow conjecture: every4-edge-connected graph admits a nowhere-zero3-flow. Group connectivity was introduced by Jaeger et al as a. generalization of nowhere-zero flows. In1992, they conjectured that every5-edge-connected graph is Z3-connected. In this paper, we do some work on these two conjectures, as follows.Firstly, we investigated the Chvatal-Erdos condition for group connectivity in graphs. If G*is the graph obtained from G by continually contracting nontrivial A-connected subgraph, then we say that G can be A-reduced to G*. Let G be a2-edge-connected simple graph and let A be an (additive) abelian group with|A|≥4. We proved that if k’(G)≥α(G)-1, then G is A-connected or G can be A-reduced to G*€{C4, C5, G1, G5, G6} or G is one of the13exceptional graphs.Secondly, we investigated the maximum degree condition and group connectiv-ity. Let G be a3-edge-connected simple graph on n>19vertices and let A be an abelian group with|A|≥4. We proved that if max{d(u), d(v)}> n/6for every uÏ… (?) E(G), then G is A-connected. This result generalizes the result by Yao, Li and Lai [Discrete Math.,310(2010)1050-1058]. Thirdly, we investigated Z3-Connectivity of Wreath Product of Graphs G and G’. Let G be a simple connected nontrivial graph. We proved two interesting results as follows:If G’is a triangularly connected simple graph with|V(G)|>3, we get that if G’(?){K2,t/+,G1}, then GOG’is Z3-connected. Meanwhile, we also get that GpG’ is Z3-connected. If G’ is a symmetric graph, we get that if G’(?){K2,#1,2}, then GpG’ is Z3-connected.Finally, we investigated nowhere-Zero3-Flows in Semistrong Product of Graphs. It is proved that if G and H are two nontrivial connected simple graphs, then G·H admits a nowhere-zero3-flow. This result extends the study of nowhere-zero flows on product graphs, initiated by Imrich and Skrekovski, advanced by Shu and Zhang, by Rollova and Skoviera, and by others.