Cycle Graph is Bipartite iff Order is Even/Examples/C 4

Example of Use of Cycle Graph is Bipartite iff Order is Even

The cycle graph of order $4$ is the complete bipartite graph $K_{2, 2}$.

Proof

From Cycle Graph is Bipartite iff Order is Even, $C_4$ is a bipartite graph, consisting of the cycle $v_1 v_2 v_3 v_4 v_1$.

We have that $C_4$ is partitioned into two sets $A$ and $B$ such that:

$A = \set {v_1, v_3}$

$B = \set {v_2, v_4}$

and it is seen by inspection that every element of $A$ is adjacent to every element of $B$.

The result follows by definition of complete bipartite graph.

$\blacksquare$