TOGETHER
Their relationship starts out by a slap to the cheek. Wolfram's cheek to be precise and Yuuri doing the slapping. Which resulted in them being engaged for the rest of the show. A lot of people think that Yuuri slapping Wolfram was really girly of him but in the light novels Yuuri explains he slapped him instead of punching him because his face was too pretty to actually damage.
Wolfram: When Wolfram first meets Yuuri he doesn't seem to like him at
all. The reason for this is because Yuuri is half human and Wolfram hates humans.
Other than that, there are no personal feelings towards Yuuri. He doesn't want him
to be the king and says tons of rude things and even goes so far as to insult 
Yuuri's mother, whom he has never met. You might think, he is a huge brat for all
of this, but in truth, he probably wouldn't have gone that far if it wasn't for the
perfume that Wolfram's mother left in the bathroom that Yuuri used. It makes anyones emotions about the person intensify and since Wolfram wasn't happy about him being king, it came out as a hate for Yuuri.
But, after Wolfram challenges Yuuri to a duel and loses he gets a new sense of respect for Yuuri and starts to take his role as fiancée very important. He goes around shouting that Yuuri is a wimp (hannachoko) and always accuses him of cheating. Yet, he shows deep concern and caring for Yuuri by always trying to protect him, getting worried, and always wanting to be around him even though he uses the excuse of Yuuri being tempted to cheat on him if he isn't around.
When ever Yuuri goes missing he goes out to look for him and his manner is completely different because he is so worried. He isn't as hot headed, he doesn't even have time to get motion sick and all his thoughts are directed at Yuuri in hopes of finding him quickly. Wolfram even gets a bond to Yuuri that is so strong that he can just tell when it is Yuuri doing something and he knows where to be right away to help him.
Yuuri: When Yuuri first sees Wolfram he is struck with awe by Wolfram's looks. Calling him a bishounen (pretty boy) in his mind. But it doesn't take long for Yuuri to change his whole perspective on Wolfram as soon as he opens his mouth and starts to insult him. At first Yuuri doesn't care at all and doesn't even take offence to any of it until Wolfram has to say something about his mother.
Though Yuuri has a more forgiving and easy going personality than Wolfram, it seems to take Yuuri a little longer to actually get closer to Wolfram and care about him than it does for Wolfram too. His ways of showing how much he cares a little less subtle though since Yuuri says he doesn't like guys and doesn't want to marry one. He doesn't seem to take the Fiancée stuff that Wolfram always throws at him seriously, or he just doesn't want to even think about it. In time though, Yuuri comes to really enjoy Wolfram's presence and is even super relieved and happy to have Wolfram around.
Like Wolfram is always trying to protect Yuuri, Yuuri is also trying to protect Wolfram, but it's not as obvious. I don't think he always realizes he is doing it. So many times you see Yuuri change into his 'Demon' side just so he can save Wolfram. The first noticeable time would be when they are running away from lava and fire when Wolfram all of a sudden runs out to save a child and Yuuri gets so frightened for Wolfram that he changes and puts out all the fire. Another good time is when they are about to be covered in snow and Yuuri saves them in his other form and even carries Wolfram! Towards the end of the show Yuuri finally breaks down and admits to his engagment to Wolfram. He does so when Wolfram is in trouble and he wants to save him, he is talking to Conrad and says something like, I will save Wolfram no matter what, onichan (brother). Him calling Conrad, brother, was saying he was not against the engagment anymore and Conrad smiles at Yuuri in understanding.
All in all you can just tell that they each care a lot for one another and even if it isn't love, it's the closet to a main romantic relationship in this series.