1. Combinatorics of colouring 3-regular trees
Paul Sant1 and Alan Gibbons2
1 Department of Computing and Information Systems, University of Luton
2 Department of Computer Science, King’s College, London

2. Overview
Introduction – What is Colouring Pairs of Binary Trees (CPBT) ?
Rotations and Colour constraining edges
Optimal Linear-time algorithmics ?
Conclusions

3. CPBT – Problem Definition
Given any pair of same-sized 3-regular trees, T1 and T2, 3-edge colour T1 and T2 in such a way that edges adjacent to leaves (from left to right) are similarly coloured in T1 and T2 T1 T2
x1
x2
x3
x4
x5 y1 y2 y3 y4 y5 x6 y6

4. How to solve CPBT ?
We have approached this in two ways (several papers:TCS, Texts in Algorithmics, LNCS):-
Rotations coupled with colour-constrained edges
A colouring topology approach
This talk is concerned with the first of these approaches.

5. Rotations in binary trees
Well known technique used for rebalancing Binary (particularly AVL) trees :-
Any binary tree is transformable into any other using a sequence of rotations.

6. Colouring constrained edges

The rotated edge is constrained to be the same colour as its grandparent :-
It is easy to show that this condition is necessary to obtain a solution to CPBT.

7. A solution to CPBT
An example of a sequence with no clashes (colouring requires O(n) time)
As we will see, changing the rotational
path slightly may cause a clash.

8. Shortest clashing path
Example of a (shortest) clashing path :-
The rotational space of binary tree is large
finding a non-clashing path is still an open
problem

9. Non-clashing paths
The number of trees with n leaves is given by n-1th Catalan number – this is factorial in n.
Therefore, finding non-clashing paths by exhaustive search is not an option for polynomial time algorithmics.

10. Shortest clashing paths
For each n > 5 there is at least one pair of trees for which a shortest rotational path will lead to a clashing path :- {
k-1 {
k-1 {
k-1
k-2 steps
5 steps

11. Longer paths
However, we can solve all the problematic pairs previously discussed :-
And we only require one extra rotation {
k-1 {
k-1 {
k-1
k-2 steps
6 steps

12. Open problems How to find non-clashing paths ?
Can we find such non-clashing paths quickly ?
Conversely, can we prove that the problem is hard ?

13. Conclusions We postulate that the rotational approach can solve CPBT.
A linear-time solution to finding non-clashing paths would lead to a linear-time solution to 4CP for planar graphs.
Work on the problem continues…