Wikipedia:Reference desk/Archives/Mathematics/2009 May 30

= May 30 =

Eulerian city?
Does the road network in any known city in the world have an Eulerian circuit? Neon Merlin  01:03, 30 May 2009 (UTC)


 * If you ignore the roads that lead out of town, Monowi, Nebraska has an Eulerian circuit, according to this PDF street map by the Census Bureau. The Nebraska Department of Roads map, though, seems to indicate that it doesn't. —Bkell (talk) 02:28, 30 May 2009 (UTC)


 * Actually it has Eulerian path, but not a circuit. Grand and Broad streets form a junction which is a graph vertex of order 3, simillary do Grand and Marion St. so there can't be an Eulerian circuit. --CiaPan (talk) 05:46, 31 May 2009 (UTC)


 * Yes, you're right. My mistake. —Bkell (talk) 21:51, 31 May 2009 (UTC)


 * Somewhere in this world, there may be a small island with a single road running its circumference. That would qualify.  Beyond this trivial case, such a town would likely need to have a uniform grid design internally (so that every intersection has four approaches).  However, such layouts would likely have a perimeter road, with many three way intersections, or no perimeter road, with many dead-end streets at its perimeter.  Both situations would disqualify the town.  I can image a grid layout with a stairstep-like perimiter that would qualify.  Consider a town with five city blocks arranged in a plus-sign configuration, or a 1-3-1 pattern.  That would qualify, as would similar layouts of 1-3-5-3-1 or 1-3-1-3-1.  There is a remote possibility that such a town might exist.  Other layouts such as a pentagram or File:Complete graph K5.svg would qualify, but such layouts would likely not be practical.  -- Tcncv (talk) 01:38, 1 June 2009 (UTC)