MATH 443
Assignment #4
Due Thursday November 28.
1. a) Let G be a graph with even degrees. Show that G can be decomposed into even length
closed trails and a set of vertex disjoint odd cycles.
b) Let G be a graph on n vertices with the property that every pair of vertex disjoint odd
cycles
Pn are joined by an edge. Let f = (f1 , f2 , . . . , fn ) be a vector of positive integers with
i=1 fi being even. Show that G has a fractional f -factor if and only if G has an f -factor.
Hint: Obviously the half edges are the concern. But an edge joining the two odd cycles may
just be an edge of G and not a half edge of the fractional f-factor.
2. Let G be a graph
graph
H
of girth at least 6 with χ(G) = k and n(G) = n. Form a new[kn]
kn
consisting of n vertex disjoint copies of G each indexed by a unique element of n and a
set S of kn new vertices. For each subset of S of size n, join it to the corresponding copy of G
by a matching (|S| = n = |V (G)|). Show that the resulting graph has girth 6 and chromatic
number at least k + 1. Hint: If H has χ(H) = k, then there is a subset of S of size n all of
whose vertices receive the same colour. Consider the corresponding copy of G. When you are
done the construction you have a construction of a graph G with girth 6 and χ(G) ≥ k.
3. The following problem explores a vector space of a connected graph G consisting of spanning
subgraphs of G which have all even degrees (sometimes called even subgraphs). Addition in
this vector space is modulo 2 sum (our symmetric difference) and scalar multiplication is over
the field of 2 elements (the field formed by 0,1 with 1+1=0 and all other operations as you
would expect; one can think of 0 as ‘even’ and 1 as ‘odd’). Thus the spanning graph of no
edges is the zero vector in this vector space. Select a spanning tree T of G. We wish to show
that
C = {Ce : e ∈ E(G)\E(T ), Ce is the unique cycle in T + e}
is a basis for the vector space V (known as the cycle space).
a) Show that the cycles in C are linearly independent.
b) Show that if C1 , C2 are cycles (not necessarily from C) with the property that C1 \E(T ) =
C2 \E(T ), then C1 = C2 . (try symmetric difference)
c) Show that any cycle C in G is a unique linear combination of cycles in C. Here we are
using the sum of subgraphs is modulo 2 sum or our symmetric difference.
d) Show that the dimension of the cycle space, namely the vector space consisting of all
subgraphs of all even degrees, has dimension e(G) − n(G) + 1.
4. Given an (undirected) connected graph G = (V, E), we could ask for the minimum length
odd cycle. Imagine we had an algorithm that finds a minimum total weight perfect matching
(yes, such algorithms exist). Consider the following graph Gxy which you can envision as a
copy of G\y on vertices V \y and a copy of G\x on vertices V 0 \x (V 0 is just a copy of V )
where we add edges zz 0 for z ∈ V \{x, y}. Gxy . Let V 0 = {v 0 : v ∈ V }:
Gxy = (V \y) ∪ (V 0 \x0 ), E 0 )
where E 0 = {(i, i0 ) : i ∈ V \(x, y)}∪{i, j) : (i, j) ∈ E, i, j 6= y}∪{(i0 , j 0 ) : (i, j) ∈ E, i, j 6= x}
We give weights 0 to the edges zz 0 and weights 1 to the remaining edges. Show that a
minimum total weight perfect matching in Gxy is equivalent to finding a minimum weight
path from x to y that uses an even number of edges. Now use this to find a odd cycle of
minimum total weight in G.