Math 184A, Fall 2009, Prof. Tesler
Homework #8, Due Thursday November 19, 2009
Chapter 9# 1, 12, 23, 29
Chapter 10# 10∗,22
∗
,29
and the problems below: H-26 through H-30. Be sure to read the second page as well.
∗Notes:
•For all homework and test problems, unless otherwise noted, use these rules:
–Awalk allows reusing edges and vertices. (It doesn’t require it, it just allows it.)
–Inatrail, edges may not be reused but there are no restrictions on reusing vertices.
–In a path, vertices and edges may not be reused beyond the minimum amount required for
vertices: the second vertex of one edge is the first vertex of the next edge, and if it’s a closed
path, the first and last vertices are the same.
Although the book’s definition of “walk” on p. 185 says “distinct edges,” it’s not the standard
definition and the book actually allows reusing edges later on. The errata page on the book’s
website fixes this.
•Chapter 10# 10: The book’s solution is not correct, so ignore it.
•Chapter 10# 22: The length of a path or walk e1,e
2
,...,e
kis the number of edges in the sequence,
k. If you specify it using vertices, there will be k+ 1 vertices but the length is still k. E.g., a path
of length 2 from vertex ato bmay be given as a sequence of two edges {a, c},{c, b}or as a sequence
of three vertices a, c, b.
Longest path means the length is the largest possible length among all paths in the graph. If
there are two or more longest paths, they are in a tie for the longest length.
Problem H-26.
(a) How many simple graphs are there on vertex set [n]?
(b) How many simple graphs on vertex set [n] have exactly kedges?
Problem H-27. In each part below, information about a graph is given. Decide if a graph, without loops,
exists that satisfies the specified conditions or not. Justify your answers.
(a) A simple graph with degree sequence (1,1,2,3,3,5) (that is, six vertices, where two vertices have
degree 1, one has degree 2, two have degree 3, and one has degree 5).
(b) A multigraph with degree sequence (1,2,2,3,3,5).
(c) A simple graph with degree sequence (1,2,2,3,3,5).
(d) A simple graph with degree sequence (3,3,3,3).
(e) A tree with six vertices and six edges.
(f) A tree with three or more vertices, where two vertices have degree 1 and the rest have degrees ≥3.
(g) A disconnected simple graph with 10 vertices, 8 edges, and a cycle.
Problem H-28. In (a–b), let Cbe the set of courses at UCSD and Sbe the set of students. Let V=C∪S
and let {s, c}∈Eif and only if student sis enrolled in course c.
(a) Prove that G=(V, E) is a simple graph.
(b) Prove that every cycle of Ghas an even number of edges.
(c) S={Alice,Bob,Cindy,Dan,Emily}is a set of students.
C={Math 20A,Math 20B,Math 20C,Math 20D,Math 20E,Math 20F}is a set of classes.
A graph is formed in which {s, c}is an edge iff student sis enrolled in class c.
In a given quarter, a student can only be enrolled in at most one of Math 20A,B,C,D because
each is a prerequisite for the next one. However, a student who has already passed Math 20C can
take Math 20D,E,F in any order, and can even take two or three of them simultaneously. (Although
not recommended, it’s not forbidden.)
