There is currently little physics education literature examining thinking and learning in graduate education and even less literature characterizing problem solving among physics graduate students despite this being an essential professional skill for physicists. Given reports of discrepancies between physics problem solving in the undergraduate classroom and "real-world" problem solving, we sought to investigate whether this discrepancy exists at the graduate level. We first investigate the problem-solving skills present in first-year graduate physics assignments. A recent framework that characterizes problem solving as decisions-to-be-made was used. Assignments were taken from the four core courses of one academic year at one research-intensive university and coded by two researchers. We found that only 4 of the 29 decisions in the framework were present in most of the assignments. We then interviewed 11 instructors from 3 universities and asked which decisions they expected of first-year graduate students. Eleven decisions were expected by eight or more of the participants, but only four of these decisions were commonly practiced on assignments. Therefore, there seems to be a mismatch between instructor expectations and practice of problem solving on assignments. This suggests that graduate physics courses may not be aligned with the problem-solving skills that physics graduate students will need in their research or future careers.