Which of the claims below is not equivalent to the rest?
1) Every cycle in a graph "B" has an even length
2) Graph "B" is bipartite
3) Graph "B" has two components that are connected.
4) Graph "B" is two-colorable.
Which of the claims below is not equivalent to the rest?
1) Graph "B" is connected and contains no cycles
2) Graph "B" is a tree
3) The maximum degree of "B" is 2.
4) In Graph "B" there is a unique simple path between any 2-vertices.
For part (a) Konig's Theorem states that a graph is bipartite if and only if all any cycles present are of even length.
Note that bipartite graphs need not be connected. Consider a graph $G(V, E)$ with $E = \emptyset$. Bipartite and two-colorability are equivalent. I've edited to fix my mistake.
For part (b), consider a star. Is that a tree? The rest of those follow from the definition of a tree.