Solution

The correct answer is option 3.
Key Points

• Any tree is 2-colourable. True every tree is no loops and a bipartite graph with chromatic number 2. Odd levels coloured with one colour and even levels coloured with another colour.
• A graph G has no cycles of even length if it is bipartite. False

• A graph G is 2-colourable if is bipartite True. A graph K(2,3), set of  V₁ vertices coloured with one colour and set of V₂ vertices coloured with another colour.
• A graph G can be coloured with d+1 colours if d is the maximum degree of any vertex in the graph G.True as the vertex of degree d means it is adjacent to d vertices hence need d+1 different colours required. 
• A graph G can be coloured with O(log∣v∣) colours if it has O(∣v∣) degrees False as Kn has O(n^2) edges with chromatic number n so it needs O (√ V) colours 

∴ Hence the correct answer is (B) and (E) are incorrect.