Publikace na Matematicko-fyzikální fakulta |
2019
Abstrakt
Since planar triangle-free graphs are 3-colourable, such a graph with n vertices has an independent set of size at least nDIVISION SLASH3. We prove that unless the graph contains a certain obstruction, its independence number is at least nDIVISION SLASH(3-ε) for some fixed ε>0.
We also provide a reduction rule for this obstruction, which enables us to transform any plane triangle-free graph G into a plane triangle-free graph G' such that α(G')-|G'|DIVISION SLASH3=α(G)-|G|DIVISION SLASH3 and|G'|<=(α(G)-|G|DIVISION SLASH3)DIVISION SLASHε. We derive a number of algorithmic consequences as well as a structural description of n-vertex plane triangle-free graphs whose independence number is close to nDIVISION SLASH3.