Math776 Graph Theory Midterm Exam
October 25, 2013
1. (10 points) State Halls marriage theorem.
Solution: A bipartite graph G = (A, B, E ) has a matching of size |A| if
and only if |N (S )| |S | for any S A.
2. (10 points) State Tuttes two theorems on cha
Math776: Graph Theory (I)
Fall, 2013
Homework 5 Solutions
1. [page 84, #18 ] Let k 2. Show that every k -connected graph of order at
least 2k contains a cycle of length at least 2k .
Solution by Nicholas Stier: Let k 2 and let G be a k -connected
graph wi
Math776: Graph Theory (I)
Fall, 2013
Homework 6, Solution
1. [page 112, #20 ] Show that adding a new edge to
a maximal planar graph of order at least 6 always
produces both a T K5 and a T K3,3 subgraph.
2. [page 112, #22 ] A graph is called outplanar if i
Math776: Graph Theory (I)
Fall, 2013
Homework 4, solutions
1. [page 54, #11 ] Let G be a bipartite graph with bipartition cfw_A, B . Assume that (G) 1, and that d(a) d(b) for every edge ab with a A.
Show that G contains a matching of A.
Solution by James
Math776: Graph Theory (I)
Fall, 2013
Homework 3 solution
Select any 5 problems to solve. The total score of this homework is 10 points.
You get a bonus point if you solve all 6 problems correctly. You also get another
bonus point if your solution is selec
Math776: Graph Theory (I)
Fall, 2013
Homework 1 solution
1. [page 30, #2 ] Determine the average degree, number of edges, diameter, girth, and circumference of the hypercube graph Qd .
Solution by Shuliang Bai:
The average degree is d, according to its d
Math776: Graph Theory (I)
Fall, 2013
Homework 2 solution
Select any 5 problems to solve. The total score of this homework is 10 points.
You get a bonus point if you solve all 6 problems correctly. You also get another
bonus point if your solution is selec