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.