Strong chromatic index and Hadwiger number

Abstract

AbstractWe 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 that any ‐minor‐free multigraph of maximum degree has strong chromatic index at most . We present a construction certifying that if true the conjecture is asymptotically sharp as . In support of the conjecture, we show it in the case and prove the statement for strong clique number in place of strong chromatic index. By contrast, we make a basic observation that for ‐minor‐free simple graphs, the problem of strong edge‐colouring is “between” Hadwiger's Conjecture and its fractional relaxation. For , we also show that ‐minor‐free multigraphs of edge‐diameter at most 2 have strong clique number at most .