The class of graphs G defined as: an arbitrary instance of InSched with a set X of intervals. The aforementioned polynomial$-$time reduction constructs a graph G$($X$)$ based on X$.$ Let G be the class of graphs constructed this way $($one for each possible such set X$).$

Consider the min$-$Colouring problem on this class of graphs:

Given as input a graph G$,$ find a colouring of G that uses the smallest

number of colours possible.

A colouring of a graph G is a function f that assigns a colour to each node of the graph, such that for every two nodes u$,$ v with $($u$,$ v$)$ in E$,$ we have f$($u$)($not equal$)$ f$($v$).$

Is the min$-$Colouring problem on graphs in G NP$-$hard or polynomial$-$time solvable $($assuming P $($not equal$)$ NP it cannot be both$).$

let A and B be two problems that are computationally equivalent if A is polynomial$-$time reducible to B and vive$-$versa.

$-$ If we claim that it is NP$-$hard, how can I construct a polynomial$-$time reduction from an NP$-$hard problem A$.$

$-$ If we claim that it is polynomial$-$time solvable, how can I construct a polynomial$-$time reduction to a polynomially$-$time solvable problem B$.$

Please list any helpful source to go through the content of this question; any proofs, book materials or research papers. I would like to go through more about this topic.

Someone answered this but I just wanna confirm if its not Polynomial$-$time solvable and indeed NP$-$Hard because I am confused as$...$ For the class of graphs G defined, which are derived from interval scheduling instances, the graphs are essentially interval graphs... So the min$-$Colouring problem for this specific class is polynomial$-$time solvable? Thanks