Generalizations of Hadwiger’s Conjecture

Conjectured generalizations of Hadwiger’s Conjecture are discussed. Among other results, it is proved that if \(X\) is a set of \(1\), \(2\) or \(3\) vertices in a graph \(G\) that does not have \(K_6\) as a subcontraction, then \(G\) has an induced subgraph that is \(2\)-, \(3\)- or \(4\)-colourable, respectively, and contains \(X\) and at least a quarter, a third or a half, respectively, of the remaining vertices of \(G\). These fractions are best possible.