The short answer is yes. However, there are certain guidelines you need to follow in order to make your experience of the trip smooth.
1. Have a bicycle bag or a cover. It doesn’t matter if it’s a fancy one, costing a hundred Euros, or a piece of film for a few Euros. You just need to have it in order to comply with the railway rules.
2. Remove the font wheel before taking the bike on a train. You can use the cover before getting on a train, or when inside (if it’s easier). But the wheel should be removed first, so that the train staff is satisfied. This is because, the rules say that the bicycle must be disassembled for taking on a train. Removing the front wheel is usually enough.
3. When inside the train, use the luggage shelf at the very top for storing the bike. In fact, it’s more secure when you put the bike across two shelves of different sides in the cabin. Alternatively, you can put the bike in the smoking area at the end of the carriage. Ask permission from a member of staff first, and use a locker to security.
The last, but not the least. Be friendly with the staff, even if they are not. Although, developing fast, the bicycle culture is not widespread in Ukraine yet, so not many people are taking bicycles on the trains. So, the train staff sometimes may be not confident how to handle the passengers with the bikes. Help them, by pointing out that the rules of the Ukrainian Railway say that any passenger who got a train ticket is allowed to take a bicycle on a train with him free of charge, as long as it’s dissembled and has a bike cover.