Yeah, if Tex simply puts it to full auto (canoncially, it's a selective-fire weapon, from semi-auto, 3-burst to full auto) and fire in controlled bursts, or simply uses the burst fire, I'd put my money on Tex.
If it comes to melee, would the whole being a SPARTAN help? I mean, they've got SOOPER reflexes and shit, not to mention years of training since childhood to be cold-stonehearted killers...
Well, with AI help the single luckiest SPARTAN ever (that's actually a canon fact) managed to deflect a missile with his bare hand. So Tex would stand a chance in melee, just not a great one considering the instant-kill nature of Luke's lightsaber and his Force sense.
However, one thing we're not really discussing is the prep time. Tex is a disciplined soldier, and has shown to be really good at thinking up traps and tactics to win battles she normally couldn't. She'd probably spend a lot of time (3-5 hours) thinking specifically of how Luke's combat style works, and how to defeat him. So you never know what she might come up with.
@Barbarossa: An ordinary man has no chance in hell of wrestling open Ajax's arms. And this is the first time the weak neck has been mentioned. And if his neck is a weak-point, Mustang more or less insta-wins with the first flame. Also, logic alert.
If Ajax's neck is susceptible, shouldn't his throat be as well? Thus, he should have at the very least gained a horrid, incurable cough from being in that fire. Considering he did not, I'm not seeing how it's all that important a detail.