Weight Choosability of theta Graphs

The 1,2,3-conjecture is a problem of edge weight colorability of graphs which was posed by M. Karoński et al in 2004.
Further problem of edge weight choosability of graphs was posed by T. Bartnicki et al in 2009.
While being solved for some special cases, the two problems are still open nowadays.
In this thesis, we use the combinatorial nullstellensatz and the permanent to find some results.
We go through the cycles, then discuss the θ-graphs and generalized θ-graphs.
The main result of this thesis is to show these graphs are all 3-edge weight choosable.