MATH20902: Discrete Maths, Problem Set 5
These problems, are also available from http://bit.ly/DiscreteMancMaths. Some
refer to Dieter Jungnickel’s book, Graphs, Networks and Algorithms, an online copy
of which is available from within the university at http://bit.ly/Jungnickel2.
More about Eulerian graphs
In addition to the exercises below, we’ll explore an algorithm to find Eulerian tours.
(1) (after Jungnickel’s Exercise 1.3.3)
Let G be a connected multigraph with exactly 2k vertices of odd degree (and k 6= 0).
Show that the edge set of G can be partitioned into k trails.
(2) (after Jungnickel’s Exercise 1.3.4)
The line graph L(G) has as its vertices the edges of G. Two vertices in L(G) are
adjacent if and only if the corresponding edges in G are incident on a common
vertex.
(a) Show that the line graph L(Kn ) of the complete graph Kn is isomorphic to
the triangle graph Tn from Problem Set 1.
(b) Obtain a formula that relates the degree of a vertex in L(G) to the degrees of
the endpoints of the corresponding edge in G.
(3) (after Jungnickel’s Exercise 1.3.5)
Let G be a connected graph. Find a necessary and sufficient condition for L(G) to
be Eulerian. Conclude that the line graph of an Eulerian graph is also Eulerian and
show that the converse need not be true (that is, find a counterexample).
(4) Suppose that a connected multigraph G is known to be Eulerian (that is, all
its vertices have even degree): show that the following algorithm will construct an
Eulerian tour.
Hierholzer’s Algorithm
Takes an n-vertex, connected multigraph G in which deg vj is even for all vertices
vj and finds an Eulerian tour K. We will assume that the edges are numbered E =
{e1 , . . . , em } and will describe the tour as a sequence of edges K = (ei1 , ei2 , . . . , eim ).
We will construct a sequence of tours C0 , C1 . . . Ck = K.
(1) Set j ← 0 and w ← v1 . Note that all the edges are currently unused.
(2) Construct a closed trail Zj by following the lowest numbered unused edge that
is incident on w to some new vertex u. Then take the lowest-numbered unused
edge that is incident on u and follow it to a new vertex, say, u0 . . . . Continue
in this way until you return to w. (Why are you sure to return?)
(3) If j = 0, say C0 = Z0 . Otherwise, form a new tour Cj that starts at w, traces
over the edges of Cj−1 and then traces over those of Zj .
(4) If Cj includes every edge in the graph, stop: K = Cj . If not, set j → j + 1
and go to step 5.
(5) Find the lowest numbered vertex that (i) appears in Cj−1 and (ii) has some
unused edges incident on it: make this vertex the new w and go to step 2.
(This is the heart of the algorithm: why is it true that if unused edges remain,
some of them must be incident on a vertex that’s part of Cj−1 ?).
More about Hamiltonian graphs
(5) We begin with a few easy exercises
(a) For which values of d is the cube graph Id different from its closure [Id ]?
(b) For which values of d is the cube graph Eulerian?
(c) For which values of d is the cube graph Hamiltonian?
(d) Find a Hamiltonian cycle in the graph pictured below.
(6) Consider a graph G(V, E) and prove that if degG (v) < 2, then v has the same
degree in both G and [G]. That is, the closure construction never adds an edge to
a vertex v with degree less than two.
(7) (after Jungnickel’s Exercise 1.4.5)
Find the minimal number of edges needed to make a graph G on six vertices whose
closure [G] is the complete graph K6 .
(8) (after Jungnickel’s Exercise 1.4.6)
Show that if G is Eulerian then L(G) is Hamiltonian. Does the converse hold? If
so, prove it: if not find a counterexample.