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

- for $i=1,.., k$ every $v_i$ is contained in a path,
- each path consists of vertices only from $V_k$ except for exactly one of its endpoints which must be outside of $V_k$,
- in each path the red and blue edges are alternating?

The claim is true if $k$ is small (<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.

**Update 2015.06.26** As there is renewed interest, let me add that here is the since then published paper: http://arxiv.org/abs/1104.0642 and last year the claim for which they needed this lemma was solved in another way by Gyarfas: http://www.degruyter.com/view/j/dmgt.2014.34.issue-1/dmgt.1735/dmgt.1735.xml.