Dog attacks other dog whenever I come home

Simple alpha challenge? Bigger dogs accepts you as the alpha and smaller does not.

Or maybe the small dog is just jealous and wants your undivided attention.

It sounds like by playing with them more and taking them for more walks you're actually on the right track, as tired dogs are happy dogs. Small breeds are definitely a bit... funny, at times, they can crave attention more, be more possessive, etc.

Generally I'd say go for the ignore option and not let them get over excited by you being home, but you'd need to ideally enforce the behaviour that you want to see. So you're correct, you will have to train them a bit.

IF your dogs are foodies you could try this: you come home, and get the dogs to sit, as soon as they come up to you. If they sit, give them a treat. Do this for both dogs. If one doesn't sit, that dog doesn't get a treat. This in theory should get the smaller dog to realise that if he comes up and sits for you as soon as you come in he gets rewarded, if he comes up and barks or bites or shows any unwanted behaviour he gets nothing - but his bigger companion is sitting and he/she is getting a treat!

Alternatively, or in addition to this, you could separate the dog for a period for the bad behaviour. Pop him somewhere else - or better yet from our dog trainers, you leave for a bit, and come back. This generally works with barking bad habits. The idea being that you want my attention, well you only get it on my terms, so behave or I'm not going to be in the room with you.

We use a similar behaviour when our dog gets too excited playing ball and starts to bite at hands. The ball gets taken away and he has to sit to earn it back again.