Safe glue

[Vierville]

I have been thinking of making a maze or tube network for my hamsters, and have saved all the toilet and kitchen roll cores and cardboard boxes I've come across for a few weeks now. I also have some icecream sticks that I may use too.

Being cardboard or wood, they will need to be stuck together with glue and I was wondering which type of glue would be safest? I'm assuming wood glue/white glue? Many glues I've looked at say 'non toxic' but I assume that is non toxic to humans, not necessarily hamsters.

Thanks!