First of all, if a graph is k colourable, that means it is also k+i colourable for any positive i
Option A is true. Every graph of $v$ vertices is $v$ colourable. Just give each vertex it's own colour.
Option B is the definition of of bipartite graph.
Bipartite $\equiv$ 2 colourable $\equiv$ no-odd-length-cycle.
Option B is true
Option D asks if we have a polynomial time algorithm to check if a graph is bipartite. Yes, we have. (DFS)
Option D is true
Option C is true. Maximum degree gives us the maximum neighbours a node can have. Each neighbour gets a different colour.
Option E is false. The Grötzsch graph is a triangle-free graph that needs four colours.