Group connectivity and group colorings of graphs . A survey

In 1950s, Tutte introduced the theory of nowhere-zero flows as a tool to investigate the coloring problem of maps, together with his most fascinating conjectures on nowhere-zero flows. These have been extended by Jaeger et al. in 1992 to group connectivity, the nonhomogeneous form of nowhere-zero flows. Let G be a 2-edge-connected undirected graph, A be an (additive) abelian group and A* = A . {0}. The graph G is A-connected if G has an orientation D(G) such that for every map b: V (G) . A satisfying .v.V(G) b(v) = 0, there is a function f: E(G) . A* such that for each vertex v . V (G), the total amount of f-values on the edges directed out from v minus the total amount of f-values on the edges directed into v is equal to b(v). The group coloring of a graph arises from the dual concept of group connectivity. There have been lots of investigations on these subjects. This survey provides a summary of researches on group connectivity and group colorings of graphs. It contains the following sections. 1. Nowhere-zero Flows and Group Connectivity of Graphs 2. Complete Families and A-reductions 3. Reductions with Edge-deletions, Vertex-deletions and Vertex-splitting 4. Group Colorings as a Dual Concept of Group Connectivity 5. Brooks Theorem, Its Variations and Dual Forms 6. Planar Graphs 7. Group Connectivity of Graphs 7.1 Highly Connected Graphs and Collapsible Graphs 7.2 Degrees Conditions 7.3 Complementary Graphs 7.4 Products of Graphs 7.5 Graphs with Diameter at Most 2 7.6 Line Graphs and Claw-Free Graphs 7.7 Triangular Graphs 7.8 Claw-decompositions and All Tutte-orientations