Victor La
MSTM 6036
23 February 2011
Homework #5
2) (a) For which values of of n does Kn , the complete graph on n vertices, have an Euler
cycle?
Recall that an undirected multigraph has an Euler cycle if and only if it is connected
and has all vertices of even valence. Thus, we want to find Kn where each vertex will be of
even valence. Given n vertices, Kn will have n(n−1)
edges. If the total number of valences
2
in Kn divided by n vertices, the result will be even, thus the Kn having an Euler cycle. We
also know the total number of valences of any graph is twice the number of edges. Thus,
X
valencesKn
n(n − 1)
=2
2
= n(n − 1)
⇒ n(n − 1) is the total number of valences ∀ Kn graph.
Now we take the total number of valences, n(n − 1) and divide it by n vertices ∀ Kn
graph and the result is n − 1. n − 1 is the valence each vertex will have in any Kn graph.
Thus, for a Kn graph to have an Euler cycle, we want n − 1 to be an even value. But we
already know in terms of complete graphs, if the number of n vertices is odd, we will have
an even n − 1 value and if n is even, we will have an odd n − 1 value. In other words,
odd Kn has an Euler cycle
For n =
even Kn does NOT have an Euler cycle
(b) Are there any Kn that have Euler trails but not Euler cycles?
Recall the corollary - A multigraph has an Euler trail, but not an Euler cycle, if and
only if it is connected and has exactly two odd-valent vertices. From the result in part
(a), we know that any Kn graph that has any odd-valent vertices, every vertex will be
odd-valent. Thus, contradicting the corollary of having exactly two odd-valent vertices.
Thus, there are not any Kn graphs that have strictly Euler trails, but not Euler cycles.
(c) For which values r and s does the complete bipartite graph Kr,s have an Euler cycle?
When r and s are strictly even values.
1
2
3) Find a graph G with 7 vertices such that G and its complement have an Euler cycle.
Graph G below is made up of one 7-cycle and G is made up of one 14-cycle, both
being Euler cycles.
3
15) Try to find a minimal set of edges in the graph below whose removal produces an Euler
cycle.
4
19) A set of 8 binary digits (0 or 1) are equally spaced about the edge of a disk. We want
to choose the digits so that they form a circular sequence in which every subsequence of
length three is different. Model this problem with a graph with 4 vertices, one for each
different subsequence of two binary digits. Make a directed edge for each subsequence of
three digits whose origin is the vertex with the first two digits of the edge’s subsequence and
whose terminus is the vertex with the last two digits of the edge’s subsequence.
(a) Build this graph.
(b) Show how an Euler cycle will correspond to the desired 8-digit circular sequence.
If we take the first digit of every vertex going through an Euler cycle, it will provide
the desired 8-digit circular sequence.
(c) Find such an 8-digit circular sequence with this graph model.
Starting with
(d) Repeat the problem for 4-digit binary sequences.