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On finding another room-partitioning of the vertices

Let T be a triangulated surface given by the list of vertex-triples of its triangles, called rooms. A room-partitioning of T is a subset R of the rooms such that each vertex of T is in exactly one room in R. We prove that if T has a room-partitioning R, then there is another room-partitioning of T which is different from R. The proof is a simple algorithm which walks from room to room, which however we show to be exponential by constructing a sequence of (planar) instances, where the algorithm walks from room to room an exponential number of times relative to the number of rooms in the instance. We unify the above theorem with Nash’s theorem stating that a 2-person game has an equilibrium, by proving a combinatorially simple common generalization.

Published in:

Electronic Notes in Discrete Mathematics, 36, 1257-1264

Presented at:

International Symposium on Combinatorial Optimization (ISCO), Hammamet, March 24-26, 2010

ISSN:

1571-0653

Laboratories:

Record created 2010-02-15, last modified 2018-09-13