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

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

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

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.

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.

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.

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.

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

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.

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

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

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 ?

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…