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
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. Defi
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
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
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 distinc
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
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) (rec
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 mea
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
isomorph
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 , .
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 distinc
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) (rec
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. Det
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 exa
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
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
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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
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 o
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-inciden
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-inciden
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 a
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 e
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