Group Connectivity and Group Coloring: Small Groups versus Large Groups

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 $|\Gamma'| \geqslant g(k)$. We prove that $g(k)$ exists and satisfies for infinitely many $k$, \begin{align*}(2-o(1)) k < g(k) \leqslant 8k^3+1.\end{align*} 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 $|\Gamma'| \geq h(k)$. Then $h(k)$ exists and satisfies for infinitely many $k$, \begin{align*}(2-o(1)) k < h(k) < (2+o(1))k \ln(k).\end{align*} 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.

Disproof of the group coloring version of the Hadwiger conjecture

Inspired by the idea of Barát, Joret and Wood for disproving the list Hadwiger conjecture [The Electronic Journal of Combinatorics 18 (2011), P232], we disprove the group coloring version of the Hadwiger conjecture, which asserts that if G is a Kk-minor-free graph, then χg(G) ≤ k, where χg(G) denotes the group chromatic number of G.

HAJÓS-LIKE THEOREM FOR GROUP COLORING

The group coloring of graphs is a new kind of graph coloring, introduced by Jaeger et al. in 1992, and the group chromatic number of a graph G is denoted by χg (G). In this note, we prove that for a positive integer k, a graph G with χg (G)>k can be obtained from any complete bipartite graph G0 with χg(G0)>k by certain types of graph operations.