## 5Spanning minor, Kotzig frames

We continue the study of weight decomposition.

Recall Definition 3.1.1 for weight decomposition. Let (*G,w*) be an eulerian weighted graph. A set of eulerian weighted subgraphs *{(G*_{i},w_{i}) : *i* = *1,…,t}* is called an eulerian weight decomposition of (*G,w*) if each *G*_{i} is a subgraph of *G* and for every *e ∈ E*(G) (where *w*_{i}(e) = 0 if *e ∉ E*(G)). (See Figure 3.1 where *t* = 2.)

In this chapter, we will further investigate how to find a set of three weighted graphs {(*G*_{1},w_{1}), (*G*_{2},w_{2}), (*G*_{3}, w_{3})} for some families of cubic graphs *G* such that

(*1) (G*_{1}, w_{1}) + (*G*_{2}, w_{2}) + (*G*_{3}, w_{3}) *is a weight decomposition of* (*G, 2*),

(*2) and each* (*G*_{j},w_{j}) ...