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| Content Provider | Springer Nature Link |
|---|---|
| Author | Zhan, Mingquan Shao, Yehong Li, Xiangwen Lai, Hong Jian |
| Copyright Year | 2011 |
| Abstract | 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 Σv∈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 Graphs2.Complete Families and A-reductions3.Reductions with Edge-deletions, Vertex-deletions and Vertex-splitting4.Group Colorings as a Dual Concept of Group Connectivity5.Brooks Theorem, Its Variations and Dual Forms6.Planar Graphs7.Group Connectivity of Graphs 7.1Highly Connected Graphs and Collapsible Graphs7.2Degrees Conditions7.3Complementary Graphs7.4Products of Graphs7.5Graphs with Diameter at Most 27.6Line Graphs and Claw-Free Graphs7.7Triangular Graphs7.8Claw-decompositions and All Tutte-orientations |
| Ending Page | 434 |
| Page Count | 30 |
| Starting Page | 405 |
| File Format | |
| ISSN | 14398516 |
| e-ISSN | 14397617 |
| Journal | Acta Mathematica Sinica, English Series |
| Issue Number | 3 |
| Volume Number | 27 |
| Language | English |
| Publisher | Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society |
| Publisher Date | 2010-09-06 |
| Publisher Place | Heidelberg |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Graphs and abstract algebra (groups, rings, fields, etc.) group connectivity number Group connectivity group coloring Mathematics Coloring of graphs and hypergraphs group chromatic number |
| Content Type | Text |
| Resource Type | Article |
| Subject | Applied Mathematics |
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