Math 314 9 Apr. 2013
1. Draw the de Bruijn graph for n = 4. Use this to find two distinct de Bruijn sequences for
n = 4 (distinct means they are not the same up to rotational equivalence).
2. Prove or disprove the following statement: If G is an Eulerian
Math 314 7 Feb. 2013
1. Is it possible to delete a vertex of degree (G) and reduce the average degree? Is it possible to
delete a vertex of degree (G) and increase the average degree? Explain.
2. Defining distance d(u, v) to be the length of the shortest
Math 314 5 Feb. 2013
1. Let G be a bipartite graph. Show that Ak will have zero entries for each value of k.
2. For each k > 3, determine the smallest n such that (a) there is a simple k-regular graph
with n vertices; (b) there are two non-isomorphic k-re
Math 314 18 Apr. 2013
1. Show that (G) = (G).
2. Show that (G)(G) > |V| (recall (G) is the independent number).
3. Show that (G + H) = (G) + (H) (recall G + H is the join of G and H).
4. Show that (GH) > maxcfw_(G), (H) (recall GH is the Cartesian product
Math 314 26 Mar. 2013
1. Show that if a connected graph has 2k vertices of odd degree (with k > 1) then it can be
decomposed into k open trails (trails where the beginning and end vertices are distinct).
2. The above result shows that the Petersen graph c
Math 314 17 Jan. 2013
In-class problems
1. [1.15] Suppose (G) = k. Prove that there exists a supergraph H of G (that is, a graph H
that contains G as a subgraph) such that G is an induced subgraph of H and H is k-regular.
2. Show that if a graph G on n ve
Math 314 23 Apr. 2013
1. Show that (G)(G) > |V| (recall (G) is the independent number).
2. Show that (G + H) = (G) + (H) (recall G + H is the join of G and H).
3. Show that (GH) > maxcfw_(G), (H) (recall GH is the Cartesian product of G and H).
4. Show `
Math 314 4 Apr. 2013
1. Can the edges of the Petersen graph be decomposed into 5 paths?
2. Draw the de Bruijn graph for n = 4. Use this to find two distinct de Bruijn sequences for
n = 4 (distinct means they are not the same up to rotational equivalence).
Math 314 24 Jan. 2013
1. Show that graphs G and H are isomorphic if and only if the graphs G and H are.
2. Give an example of two graphs which have the same (multi-)set of degrees but are not
isomorphic.
3. [1.15] Suppose (G) = k. Prove that there exists
Math 314 28 Feb. 2013
1. If a graph has fewer edges than vertices, must the graph have a component which is a
tree? Explain.
2. Show that the number of labeled trees on n vertices with degrees d1 , . . . , dn is
(n 2)!
.
(d1 1)!(d2 1)! (dn 1)!
3. In the g
Math 314 28 Mar. 2013
1. Show that if a connected graph has 2k vertices of odd degree (with k > 1) then it can be
decomposed into k open trails (trails where the beginning and end vertices are distinct).
2. The above result shows that the Petersen graph c
Math 314 25 Apr. 2013
1. Show that (G)(G) > |V| (recall (G) is the independent number).
2. Show that (G + H) = (G) + (H) (recall G + H is the join of G and H).
3. Show that (GH) > maxcfw_(G), (H) (recall GH is the Cartesian product of G and H).
4. Show `
Math 314 22 Jan. 2013
In-class problems
1. Draw a 3-regular bipartite graph that is not K3,3 .
2. If G is a k-regular graph then is G also a regular graph? If so what is the degree of a
vertex?
3. Determine the number of 7-regular graphs on 10 vertices.
4
Math 314 15 Jan. 2013
In-class problems
1. Give an example of a graph that models something (either abstract or physical).
2. Is it possible that among a group of seven people that each person has exactly three
friends in the group? Explain.
3. Determine
Math 314 21 Feb. 2013
1. Find the smallest graph with (G) = 3 having a pair of nonadjacent vertices linked by
four pairwise internally disjoint paths.
2. If a graph has fewer edges than vertices, must the graph have a component which is a
tree? Explain.
3
Math 314 12 Feb. 2013
1. Defining distance d(u, v) to be the length of the shortest path between two vertices u and
v, show that this is a metric, i.e.,
(a) d(u, v) > 0 and d(u, v) = 0 if and only if u = v.
(b) d(u, v) = d(v, u).
(c) d(u, v) + d(v, w) > d
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Math 314 2 Apr. 2013
1. Show that if a connected graph has 2k vertices of odd degree (with k > 1) then it can be
decomposed into k open trails (trails where the beginning and end vertices are distinct).
2. The above result shows that the Petersen graph ca
Math 314 11 Apr. 2013
1. Draw the de Bruijn graph for n = 4. Use this to find two distinct de Bruijn sequences for
n = 4 (distinct means they are not the same up to rotational equivalence).
2. Prove or disprove the following statement: If G is an Eulerian
Math 314 7 Mar. 2013
1. Show that the number of labeled trees on n vertices with degrees d1 , . . . , dn is
(n 2)!
.
(d1 1)!(d2 1)! (dn 1)!
2. In the graph obtained from K5 by deleting two non-incident edges, assign weights 1, 1,
2, 2, 3, 3, 4, 4 to the e
Math 314 5 Mar. 2013
1. Show that the number of labeled trees on n vertices with degrees d1 , . . . , dn is
(n 2)!
.
(d1 1)!(d2 1)! (dn 1)!
2. In the graph obtained from K5 by deleting two non-incident edges, assign weights 1, 1,
2, 2, 3, 3, 4, 4 to the e
Math 314 19 Feb. 2013
1. Find the smallest graph with (G) = 3 having a pair of nonadjacent vertices linked by
four pairwise internally disjoint paths.
2. Let G be a 2-connected and e an edge going u and v. Show that G e is 2-connected if
and only if u and
Graph theory
The first reviewining
A graph is a pair of sets G = V(G), E(G) where
V(G) = V is the set of vertices and E(G) = R is
the set of edges where an edge consists of an (unordered) set of two elements. Visually a graph is a
set of points (the verti
Math 314 26 Feb. 2013
1. Find the smallest graph with (G) = 3 having a pair of nonadjacent vertices linked by
four pairwise internally disjoint paths.
2. If a graph has fewer edges than vertices, must the graph have a component which is a
tree? Explain.
3