Group connectivity and group coloring: Small groups versus large groups

A well-known result of Tutte says that if Γ is an Abelian group and G is a graph having a nowhere-zero Γ-flow, then G has a nowhere-zero Γ^′-flow for each Abelian group Γ^′ whose order is at least the order of Γ. 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 Γ-connected for some Abelian group Γ of order k, then G is also Γ^′-connected for every Abelian group Γ^′ of order |Γ^′ | ≥ g(k). We prove that g(k) exists and satisfies for infinitely many k, (2 − o(1))k ≤ g(k) 8k³ + 1. The upper bound holds for all k. Analogously, we define h(k) as the least number such that, if G is Γ-colorable for some Abelian group Γ of order k, then G is also Γ^′-colorable for every Abelian group Γ^′ of order |Γ^′ | ≥ 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.