MATH 443
Problems #3
Due Tuesday October 15.
These problems are for classroom presentations and some fraction of them may be final exam
questions.
11. Assign integer weights to the edges of Kn . Prove that on every cycle the total weight is even
if and only if the subgraph consisting of edges with odd weight is a spanning complete bipartite
subgraph. Note the bipartition could be X = V (G), Y = ∅. Hint: Show that every component of
the subgraph consisting of the edges with even weight is a complete graph.
12. Consider a loopless G and the digraph D arising from G obtained by taking each edge e = xy in
G and creating two edges e1 , e2 with t(e1 ) = h(e2 ) = x and h(e1 ) = t(e2 ) = y. Give necessary and
sufficient condition on G so that there exists a function f : E(D) → {0, 1} satisfying f (e1 )+f (e2 ) =
1 for each pair e1 , e2 as described above and also satisfying
X
e∈E(D):h(e)=v
f (e) =
X
f (e)
e∈E(D):t(e)=v
(I think of this as flow in equals flow out).
13. Can you decompose the 15 edges of the Petersen graph into 3 paths each of 5 edges?
14. We use the notation Ka1 ,a2 ,...,an to denote the n-partite graph with parts of size a1 , a2 , . . . , an
where each pair of vertices are joined by an edge of they do not come from the same part. We see
that Ka1 ,a2 ,...,an is the complement of the graph which has n components Ka1 ∪ Ka2 ∪ · · · ∪ Kan .
Show that Ka1 ,a2 ,...,an has a perfect matching if and only if there is a loopless graph (a multigraph)
on n vertices with degree sequence a1 , a2 , . . . , an .
15. Show that for a tree T , that any automorphism φ of T has either a vertex fixed by φ i.e. some
vertex y with φ(y) = y) or there is an edge fixed by φ (an edge xy will be fixed if either φ(x) = y
and φ(y) = x or φ(x) = x and φ(y) = y).
16. The hypercube Qk is the graph with 2k vertices corresponding to the 2k bit strings (i.e. (0,1)strings) of length k where two vertices are joined if and only if the two bit strings differ in 1 position.
You might first verify that every vertex has degree k and hence there are k2k−1 edges. Show that
there is no subgraph of Qk isomorphic to K2,3 .