论文标题
小组连通性和群体着色：小组与大组
Group connectivity and group coloring: small groups versus large groups
论文作者
论文摘要
Tutte的一个众所周知的结果说，如果Gamma是一个Abelian组，而G是一个没有零伽玛 - 流的图形，那么对于每个Abelian Gampa'的gamma'的命令至少是Gamma命令的GMA'-gamma'-flow。 Jaeger，Linial，Payan和Tarsi观察到，这并没有扩展到他们更一般的团体连接概念。由此，我们将g（k）定义为最少的数字，这样，如果g对某些ABELIAN组的伽马伽玛连接，则G级的Abelian组伽玛，则G也将Gamma's连接到每个Abelian Group gamma gamma'| gamma'| > g（k）。我们证明，G（k）的存在并满足了无限的许多k，（2 -o（1））k <g（k）<= 8k^3 + 1。所有k的上限均保持。类似地，我们将h（k）定义为最少的数字，因此，如果对于某些A级的Abelian组伽玛是可以伽马色的，则G gamma'的每个Abelian Gampa'gamma'| gamma'|也可以gama'-oloRorororther gma'-colorororor。 > h（k）。然后，H（k）的存在并满足了无限的许多k，（2 -o（1））k <h（k）<（2 + o（1））k ln（k）。上限（所有k）遵循Král'，Pangrác和Voss的结果。下限是由我们在g（k）上的下限的双重性来进行的，因为平面图证明了该结合。
A well-known result of Tutte says that if Gamma is an Abelian group and G is a graph having a nowhere-zero Gamma-flow, then G has a nowhere-zero Gamma'-flow for each Abelian group Gamma' whose order is at least the order of Gamma. Jaeger, Linial, Payan, and Tarsi observed that this does not extend to their more general concept of group connectivity. Motivated by this we define g(k) as the least number such that, if G is Gamma-connected for some Abelian group Gamma of order k, then G is also Gamma'-connected for every Abelian group Gamma' of order |Gamma'| > g(k). We prove that g(k) exists and satisfies for infinitely many k, (2 - o(1))k < g(k) <= 8k^3 + 1. The upper bound holds for all k. Analogously, we define h(k) as the least number such that, if G is Gamma-colorable for some Abelian group Gamma of order k, then G is also Gamma'-colorable for every Abelian group Gamma' of order |Gamma'| > h(k). Then h(k) exists and satisfies for infinitely many k, (2 - o(1))k < h(k) < (2 + o(1))k ln(k). The upper bound (for all k) follows from a result of Král', Pangrác, and Voss. The lower bound follows by duality from our lower bound on g(k) as that bound is demonstrated by planar graphs.
