Ear decomposition of connected graph

Question

Let $G$ be a connected but not 2-connected graph. Also, $G$ does not contain even cycles as subgraphs. Show that every block $B_i$ of $G$ is isomorphic to either an odd cycle or $K_2$ using ear decompositions. Could anyone help me with this problem?

Answer 1

Gievn that your graph is connected, a block of $G$ is a is either a bridge edge (i.e. $K_2$), or a 2-connected subgraph.

Suppose it is not $K_2$, then it's 2-connected. Therefore it admits an ear decomposition. The start cycle of the ear decomposition has to be an odd cycle $C$ because $G$ has no even cycle subgraphs.

Now try to add the first ear to $C$. Suppose your ear is incident to $C$ at $\{c_i,c_j\}$. Because $C$ is an odd cycle, the paths $\{c_i,c_{i+1},\ldots,c_j\}$ and $\{c_j,c_{j+1},\ldots,c_n,c_1,\ldots,c_i\}$ have different parity, one is even and the other is odd (because the sum has to be odd). Therefore the ear together with one of this path will create an even cycle. you cannot add any ear, therefore your block is simply an odd cycle.