# Hadwiger number of total graph

Let $G=(V,E)$ be a finite, simple, undirected graph, and let $T(G)$ be its total graph. The Hadwiger number $\eta(G)$ is the maximum $n\in\mathbb{N}$ such that $K_n$ is a minor of $G$.

Is there an example of a graph $G$ such that $\eta(T(G)) > \Delta(G) + 2$ (where $\Delta(G)$ is the maximum degree)?

• Isn't this asking about Hadwiger conjecture for total graphs? en.wikipedia.org/wiki/Hadwiger_conjecture_%28graph_theory%29 – joro May 27 '16 at 12:33
• It is easy to construct graphs of maximum degree three with arbitrarily large clique minors, so examples are easy to find even without looking at the extra edges in the total graph. – Ben Barber May 27 '16 at 16:03

The graph $T(G)$ contains as a subgraph $L(G)$, the line graph of $G$.
Abrams and I constructed a $K_{9k + 1}$ minor in the graph $L(K_{5k+2})$.