yEd 3.13 does not require a pre-installed Java 6 runtime. The error message is actually erroneous.
This a is fairly frequent Java/Yosemite problem that affects applications other than yEd as well. (See Java SE 6 on Yosemite? and Eclipse Kepler for OS X Mavericks request Java SE 6 and yosemite compatibility, now or soon? for additional information.)
What version(s) of Java do you have installed on your Mac?