Group connectivity and group coloring: small groups versus large groups

TL;DR

This paper investigates the relationship between group connectivity and coloring in graphs, establishing bounds on the minimal group order needed for certain properties to hold, with implications for graph theory and combinatorics.

Contribution

It introduces the functions g(k) and h(k) to quantify minimal group sizes ensuring connectivity and coloring properties, providing new bounds and demonstrating their existence for infinitely many k.

Findings

g(k) exists and satisfies (2 - o(1))k < g(k) <= 8k^3 + 1 for infinitely many k

h(k) exists and satisfies (2 - o(1))k < h(k) < (2 + o(1))k ln(k) for infinitely many k

Upper bounds are established for all k, with lower bounds demonstrated via planar graphs

Abstract

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 Kr\'al', Pangr\'ac, and Voss. The lower bound follows by duality from our lower bound on g(k) as that bound is demonstrated by planar graphs.