Building bridges between graph colouring, extremal and structural graph theory

The goal of this project is to make progress on and connect different important subareas of graph theory.
To this end we will focus on two research lines, both involving an in-depth analysis of extremal examples.
The main research line is about
I. Hadwiger’s conjecture and offsprings
Hadwiger’s conjecture is one of the most important conjectures in graph theory. If true, it would connect two large subareas, graph colouring and structural graph theory. 
In this project we intend to connect this conjecture with other, less well studied, conjectures and problems which are more extremal in nature.
For example, we will investigate a conjecture of Furedi, Gyarfas and Simonyi from 2005, using probabilistic and structural techniques.
This will allow to gain more insight into Hadwiger's conjecture and related problems.

As a second research line, we would like to completely settle a problem that has been solved asymptotically by the applicant.
II. Plesnik’s Problem Precise
The fundamental problem of Plesnik deals with some of the most basic concepts of a graph: order, diameter and total distance.
The part of the mathematical community that worked on this problem and related ones, is not completely satisfied with the asymptotic resolution and is wondering about a more precise answer.
We want to solve this issue by, among other progress, constructing an efficient polynomial algorithm that is able to compute the answer and extremal graph(s) for an important range of parameters.