Info Techinically

There are different types of colouring, such as edge colouring and face colouring but I choose to talk about proper vertex colouring:

  • Vertex: Vertices/nodes are coloured
  • Proper: Same colours cannot be adjacent (adjacent depends on the structure)

#todo Define#computer-science/algorithm/design/backtracking and#computer-science/algorithm/design/greedy versions

Problem Description(s)

  • Let be a simple graph
  • Let be a set of colours which each integer in mapped to a unique colour.
  • means the node has been mapped to
Decision Version

Is it possible to assign a colour to every node in such that no two adjacent nodes share the same colour?

Optimisation Version

Find the chromatic number of . That is, find the least amount of colours needed to admit a proper vertex colouring of , where every node is coloured and no two adjacent nodes share the same colour.

#todo Write the question description better.

Bounds

Upper Bound

Since a trivial solution is to have every node have a different colour, an upper bound for is:

Note, this trivial solution is the only solution in the case of complete graphs.

#todo By greedy colouring, is bounded by , where is the maximum node degree .

Lower Bound

something something cliques#todo

Solutions

Greedy Approach

A greedy algorithm can be used to attempt a graph colouring, by trying to colour the entire graph a certain colour, except for when it would result in two adjacent nodes having the same colour. In this case, change the colour of those nodes, and attempt to do it again for all of these problematic nodes. Repeat until no such problematic nodes exist.

A greedy graph colouring will always be correct, but (in most cases), is not optimal. Optimality heavily depends on the order of nodes traversed through when colouring the graphs.

#todo Algorithm

Applications

  • Conflict/Clashing Resolutions
    • Some people cannot be seated next to each other
    • Some subjects cannot be done at the same time

Resources