# Strong chromatic index and Hadwiger number

## TL;DR

This paper explores how forbidden clique minors influence the strong chromatic index and strong clique number in multigraphs and simple graphs, proposing conjectures and proving results for specific cases.

## Contribution

It introduces a conjecture relating forbidden clique minors to the strong chromatic index and proves it for certain cases, also examining related parameters in simple graphs.

## Key findings

- Conjecture for multigraphs with forbidden $K_k$-minors and maximum degree $elta$
- Proof of the conjecture for $k=4$ and strong clique number case
- Bound on strong clique number for $K_k$-minor-free multigraphs with edge-diameter at most 2

## Abstract

We investigate the effect of a fixed forbidden clique minor upon the strong chromatic index, both in multigraphs and in simple graphs. We conjecture for each $k\ge 4$ that any $K_k$-minor-free multigraph of maximum degree $\Delta$ has strong chromatic index at most $\frac32(k-2)\Delta$. We present a construction certifying that if true the conjecture is asymptotically sharp as $\Delta\to\infty$. In support of the conjecture, we show it in the case $k=4$ and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for $K_k$-minor-free simple graphs, the problem of strong edge-colouring is "between" Hadwiger's Conjecture and its fractional relaxation. For $k\geq5$, we also show that $K_k$-minor-free multigraphs of edge-diameter at most $2$ have strong clique number at most $(k-\frac{1}{2})\Delta$.