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 Received December 17, 2009, accepted April 8, 2010 406 Lai H. J., et al. 5. Brooks Theorem, Its Variations and Dual Forms