The general technique is to inhale through the nose, exhale through the mouth; if you breathe through the mouth, you'll breathe in a lot of unfiltered, cold / dry air, which you'll end up feeling in your windpipe all the way down to your lungs. The nose filters and pre-warms the air you breathe in.

I googled 'running breathing nose or mouth' and read articles on first page, and general consensus is that you can't take enough oxygen during intensive exercises by inhaling through nose only..