Let $G=(V,E)$ be a finite, simple, undirected graph, and let $T(G)$ be its [total graph](https://mathoverflow.net/questions/239806/total-chromatic-number-and-total-clique-number). 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)?



Hadwiger number of total graph
The graph $T(G)$ contains as a subgraph $L(G)$, the [line graph](https://en.wikipedia.org/wiki/Line_graph) of $G$.


Abrams and I [constructed](https://ajc.maths.uq.edu.au/pdf/60/ajc_v60_p314.pdf) a $K\_{9k + 1}$ minor in the graph $L(K\_{5k+2})$.


