most recent 30 from http://mathoverflow.net 2013-05-25T08:13:45Z http://mathoverflow.net/feeds/question/12408 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/12408/red-blue-alternating-paths domotorp 2010-01-20T12:32:13Z 2010-01-20T12:45:16Z

<p>Suppose we have two simple graphs on the same vertex set. We will call one of them red, the other blue. Suppose that for $i=1,..,k$ we have $deg (v_i)\ge i$ in both graphs, where $V_k={v_1,\ldots,v_k}$ is a subset of the vertices. Is it always possible to find a family of vertex disjoint paths such that</p> <ol> <li>for $i=1,.., k$ every $v_i$ is contained in a path,</li> <li>each path consists of vertices only from $V_k$ except for exactly one of its endpoints which must be outside of $V_k$,</li> <li>in each path the red and blue edges are alternating?</li> </ol> <p>The claim is true if $k$ is small (&lt;6). It is also true if the red graph and the blue graph are the same. This question was brought to my attention by a few friends who could use it in one of their papers in preparation.</p>