Rapper Gary Kang and actress Song Ji Hyo make a very appealing couple. They've kissed. They even won "Best Kiss" at the DramaFever awards.

He professed his love for her at awards ceremonies. He gets jealous whenever she interacts with anyone else. Recently, he got jealous when she kissed actor Seo Kang Joon during a game on the show "Running Man." He got jealous when she recently got a phone call.

"Who called you at this hour. I'm curious," he said. 'Did you get a boyfriend?"

A fortuneteller on "Running Man" once said that the so-called "Monday Couple" were very compatible and their marriage would be a "match made in heaven." Gary even proposed to Song Hye Kyo on television.

But while their onscreen relationship as longstanding guests on "Running Man," is fun to watch, and she calls him oppa, there is no proof that they are more than friends. And she has had a boyfriend for two years.

She may attend his concerts as part of the duo Leessang. He may have done a cameo in her last comedy, "Emergency Couple." But despite four years of shipping she has dated CEO Baek Chang Joo of her agency, CJES Entertainment, since 2011. And despite rumors that they've broken up, the couple still sees each other.

When the news of Song Ji Hyo dating CEO Baek Chang Joo was first made public two years ago, fans tweeted to say they felt bad for Gary.

"Why are people telling me they're sorry," he responded. "They don't need to be sorry and I don't need to hear it."

She obviously likes him as a friend but sometimes it seems as if he likes her more.

He has described his co-star as laid back, sweet and cute. When asked if he would date Song Ji Hyo if she were not dating someone else, he avoided giving a specific answer. He just said he was not very good at relationships. And he would not be good for her.

While she has stated that she has no immediate plans to marry and wants to enjoy her freedom, Gary did feel the need to secure their future together.

"On Monday, you and I are a couple until we die. Even if you marry, come over and sleep at my house on Mondays," he said.

It's hard to imagine her future husband would agree.

But then at the DramaFever awards Song Ji Hyo and Gary Kang also won the "Best Couple Not Meant To Be" award.