This would, of course, require one to define Communist for the purposes of this question. There is a spectrum on the political left from Communists over Socialists to Social Democrats, with different names in different countries for similar policies and similar names for different policies.
For instance, the German state of Thuringia is ruled by a coalition led by Bodo Ramelow, a member of Die Linke.
- Die Linke was created by the fusion of the Party of Democratic Socialism (PDS) and the WASG.
- The PDS is the successor of the Socialist Unity Party of Germany (SED).
- The SED was the ruling party of the GDR under Communist rule and itself the result of a fusion of the Communist Party of Germany (KPD) and the eastern Branches of the Social Democratic Party of Germany (SPD) -- a merger which happened under direct orders of the Soviet occupation administration. I think it is generally accepted that the SED was Communist in the sense of this question.
- The SPD was re-established in eastern Germany after the fall of the Communist rule, so it could be argued with some justification that those who remained in the SED/PDS/Linke after Reunification should be labeled "Communist." But what about today, a generation later? The PDS has the Communist Platform as a subgroup, but does that make the entire party Communist?
I would argue that Ramelow is not properly characterized as a Communist, but from a sufficiently non-Communist viewpoint the distinctions among the left might blurred.