For every positive integer n, we construct a Hasse diagram with n vertices and independence number O ( 3 / 4 ) . Such graphs have chromatic Ω 1 , which significantly improves the previously best-known constructions of diagrams having Θ log In addition, if also require girth at least k ⩾ 5 such most − 2 + o The proofs are based on existence point-line arrangements in plane many incidences avoids...

We investigate the effect of a fixed forbidden clique minor upon strong chromatic index, both in multigraphs and simple graphs. conjecture for each k ? 4 that any K -minor-free multigraph maximum degree ? has index at most 3 2 ( ? ) ?. present construction certifying if true is asymptotically sharp as ? ?. In support conjecture, we show it case = prove statement number place index. By contrast,...