ASSIGNMENT 2 MATH 231 Instruction: Attempt only even/odd numbered problems. Select even numbered problem if last digit of your roll no. is even otherwise attempt the odd one. 1. Prove chromatic polynomial for a cyclic graph is given by P G (C n ) = (k-1) n + (-1) n (k-1) 2. Prove that 1. for a simple connected planar graph with ? ≥ 3 ? ≤ 3? − 6 2. for a simple connected planar graph with ? ≥ 3 𝑎?? ?? ?𝑦?𝑙? ?? 𝑙???𝑡ℎ 3 ? ≤ 2? − 4 3. Prove Kuratows ki’s graphs (K 5 and K 3,3 ) are non-planar graphs. 4. Prove Euler’s formula for planar Graph s. i.e. n – e + f = 2 5. Apply the decomposition (reduction-contraction) theorem to find the chromatic polynomial for the following graph. 6. Find the maximum flow in the following flow network graph using Ford-Fulkerson method. Also verify the max. flow-min. cut theorem.