Math 275
Graph Theory
Fall 2014
MOVING AROUND
walk in G: alternating sequence of vertices and edges
W = v0 e1 v1 e2 v2 . . . v1 e v where G (ei ) = vi1 vi
for each i. Direction in which a loop is used matters. (In simple graph can just write W = v0 v1 v2
Math 275 Graph Theory Fall 2014
Solutions to problems due 14th November
X16, adapted from 7.2.17 from Wests book (2nd ed.). (a) Prove that if a simple graph G is hamiltonian then the cartesian product G K2 is also hamiltonian.
(b) The k-cube Qk is the rep
Math 275 Graph Theory Fall 2014
Solutions to problems due 3rd October
X4. For the graph shown below, nd a minimum weight spanning tree by using (a) Kruskals algorithm,
and (b) the Jarn
k-Prim algorithm, starting at a. In each case show the edges of the tr
Math 275 Graph Theory Fall 2014
Solutions to problems due 24th October
X10. In the network below, use the Ford-Fulkerson algorithm, starting from the zero ow, to nd a maximum
ow from c to g. You may augment along more than one path at each step if the pat
Math 275 Graph Theory Fall 2014
Solutions to problems due 31st October
X13. , from which we see that Given a network (D, c) and vertices x, y V (D), x = y, let D be the
subdigraph obtained from D at the end of the following process:
(1) D = D ( x + y);
(2
Math 275 Graph Theory Fall 2014
Solutions to problems due 19th September
B&M (2nd pr.) 3.3.4 (3.3.3 in 1st pr.). Let G be a graph with two distinct specied vertices x and y,
and let G + e be the graph obtained from G by the addition of a new edge joining
Math 275 Graph Theory Fall 2014
Solutions to problems due 7th November
X15, strengthened version of B&M (2nd pr.) 9.2.3. (a) Let C be a cycle of length at least three in a
connected graph G, and let S be a set of three vertices of C. Suppose that some com
Math 275 Graph Theory Fall 2014
Solution to mastery problem originally due 12th September
B&M (2nd pr.) 4.2.4. Show that the incidence matrix of a graph is totally unimodular if and only if the
graph is bipartite. [A matrix is totally unimodular if each o
Math 275 Graph Theory Fall 2014
Solution to mastery problem originally due 26th September
X1, corrected version of B&M (2nd pr.) 4.1.20. (a) Let T1 and T2 be subtrees of a tree T .
(i) Show that T1 T2 is a subtree of T if and only if T1 T2 = .
(ii) Show t
Math 275 Fall 2014- Test 1 Name: 3 o \MLom ~
- Write answers in spaces provided. The backs of pages may be used for rough work.
- Marks are shown in brackets [
- Please sign the Honor System pledge on the last page. 40
- On all questions you are expect
Math 275 Fall 2014 Test 2 Name: 3 imbue;
Write answers in spaces provided. The backs of pages may be used for rough work.
- Marks are shown in brackets [
- Please Sign the Honor System pledge on the last page. 40
- On all questions you are expected to
Math 275 Graph Theory Fall 2014
Solution to mastery problem originally due 24th October
X11, adapted from B&M (2nd pr.) 7.2.5(a). A pair of nonnegative integer vectors (p, q), where
p = (p1 , p2 , . . . , pm ), and q = (q1 , q2 , . . . , qn ), is said to
Math 275 Graph Theory Fall 2014
Solutions to problems due 26th September
B&M (2nd pr.) 3.2.3. Let G be a connected even graph. Show that:
(a) G has no cutedge,
1
(b) for any vertex v V , c(G v) 2 d(v).
Solution. [3+4=7] (a) Suppose e is a cutedge, inciden
Math 275 Graph Theory Fall 2014
Solutions to problems due 21st November
X18. Apply the bipartite maximum matching algorithm discussed in class (Egervrys Algorithm) to nd
a
a maximum matching in the bipartite graph below. [Begin with the matching consistin
Math 275 Graph Theory Fall 2014
Solution to problem due 10th October
X7. Apply Dijkstras Algorithm to the weighted digraph shown below to nd shortest paths from b to every
other vertex. At each step draw a separate copy of the graph, showing the current o
Math 275 Graph Theory Fall 2014
Solution to mastery problem originally due 15th October
X9. (a) Prove that if D is a digraph, x V (D), and f is a ow in D such that f (v) = 0 v V (D) cfw_x,
then f (x) = 0.
(b) Suppose D is a digraph and T is a subdigraph o
Math 275
Graph Theory
Fall 2014
BASIC GRAPH THEORY DEFINITIONS
If book and instructor disagree, follow instructor!
In example:
V (G) = cfw_a, b, c, d, e, f, g,
E(G) = cfw_u, v, w, x, y,
G maps a uv (i.e. G (a) = cfw_u, v =
uv), b vw, c vx, d wy, e
wy, f
Math 275 Graph Theory Fall 2014
Solutions to problems due 5th September
B&M (2nd pr.) 1.1.17. (a) Not assigned.
(b) Show that if G is disconnected, then G is connected. Is the converse true?
Solution. (b) Suppose that G is disconnected. Then V (G) can be