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

Homogeneous Finsler spaces with only one orbit of prime closed geodesics

When a closed Finsler manifold admits continuous isometric actions,estimating the number of orbits of prime closed geodesics seems a more reasonable substitution for estimating the number of prime closed geodesics.To extend the results of Duan,Long,Rademacher,Wang and others on the existence of two prime closed geodesics to the equivariant situation,we propose the question if a closed Finsler manifold has only one orbit of prime closed geodesics if and only if it is a compact rank-one Riemannian symmetric space.In this paper,we study this problem in homogeneous Finsler geometry,and get a positive answer when the dimension is even or the metric is reversible.We guess the rank inequality and the algebraic techniques in this paper may continue to play an important role for discussing our question in the non-homogeneous situation.

Solution to an Extremal Problem on Bigraphic Pairs with a Z3-connected Realization

Let S = （a1,...,am;b1,...,bn）, where a1,...,am and b1,...,bn are two nonincreasing sequences of nonnegative integers. The pair S= （a1,..., am; b1,..., bn） is said to be a bigraphic pair if there is a simple bipartite graph G = （X ∪ Y, E） such that a1,…, am and b1,..., bn are the degrees of the vertices in X and Y, respectively. Let Z3 be the cyclic group of order 3. Define σ（Z3, m, n） to be the minimum integer k such that every bigraphic pair S = （a1,..., am; b1,..., bn） with am, b ≥ 2 and σ（S） = a1 ＋ ... ＋ am ≥ k has a Z3-connected realization. For n = m, Yin [Discrete Math., 339, 2018-2026 （2016）] recently determined the values of σ（Z3,m,m） for m ≥ 4. In this paper, we completely determine the values of σ（Z3, m, n） for m ≥ n ≥4.

A Note on Z_3 -Connected Graphs with Degree Sum Condition

A graph G satisfies the Ore-condition if d（x）＋d（y） ≥ ｜V（G）｜ for any xy E（G）. Luo et al. [European J. Combin., 2008] characterized the simple Z3-connected graphs satisfying the Ore-condition. In this paper, we characterize the simple Z3-connected graphs G satisfying d（x）＋d（y） ≥ ｜V（G）｜ - 1 for any xy E（G）, which improves the results of Luo et al.