Three utilities problem: Difference between revisions

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The problem is an abstract mathematical puzzle which imposes constraints that would not exist in a practical engineering situation. It is part of the [[mathematical]] field of [[topological graph theory]] which studies the [[embedding]] of [[Graph (discrete mathematics)|graph]]s on [[surface (topology)|surface]]s. An important part of the puzzle, but one that is often not stated explicitly in informal wordings of the puzzle, is that the houses, companies, and lines must all be placed on a two-dimensional surface with the topology of a [[Plane (geometry)|plane]], and that the lines are not allowed to pass through other buildings; sometimes this is enforced by showing a drawing of the houses and companies, and asking for the connections to be drawn as lines on the same drawing. In more formal [[graph theory|graph-theoretic]] terms, the problem asks whether the [[complete bipartite graph]] <math>K_{3,3}</math> is [[planar graph|planar]].{{r|intuitive|bona}}

TheIt answeris toan the[[List puzzleof isimpossible negativepuzzles|impossible puzzle]]: it is not possible to connect all nine lines without crossing, or in mathematical terms the graph <math>K_{3,3}</math> connecting each house to each utility is not planar. Multiple proofs of this impossibility are known, and form part of the proof of [[Kuratowski's theorem]] characterizing planar graphs by two forbidden subgraphs, one of which is <math>K_{3,3}</math>. Versions of the problem on nonplanar surfaces such as a [[torus]] or [[Möbius strip]] can be solved.

<math>K_{3,3}</math> is often referred to as the '''utility graph''' in reference to the problem;{{r|gs93}} it has also been called the '''Thomsen graph''' after 19th-century chemist [[Hans Peter Jørgen Julius Thomsen|Julius Thomsen]]. It has six vertices, split into two subsets of three vertices, and nine edges, one for each of the nine ways of pairing a vertex from one subset with a vertex from the other subset. It is a [[well-covered graph]], the smallest [[triangle-free graph|triangle-free]] [[cubic graph]], and the smallest non-planar [[Laman graph|minimally rigid graph]]. Although it is nonplanar, it can be drawn with a single crossing, a fact that has been generalized in [[Turán's brick factory problem]], asking for the minimum number of crossings in drawings of other complete bipartite graphs.