## 4Small oddness

### 4.1 **k-even subgraph double covers**

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*_{i})). (See Figure 3.1 where *t* = 2.)

Weight decomposition is an approach to the circuit double cover conjecture: if one can find a weight decomposition of (*G*, 2) such that each (*G*_{i}, *w*_{i}) has a faithful circuit cover (where ...