DUE: IN CLASS WEDNESDAY 23 JULY 2014
1. Let G be a graph with subgraphs H and K such that G = H K.
Suppose G contains a subdivision of K5 .
(a) In this rst part, we assume H and K have at most one vertex in
common. Prove that one of H
DUE: IN CLASS WEDNESDAY 16 JULY 2014.
1. Determine (with some justication) which cycles in Kn and Km,n are
chordless and which are peripheral.
BONUS: do the same for the 4-dimensional cube Q4 . (Recall
that the n-dimensional cube Qn ha
DUE: IN CLASS WEDNESDAY 9 JULY 2014.
1. (a) Let G be a 2-connected graph and let H be a subdivision of G.
Prove that H is 2-connected and deduce that, for any two edges e and
f of G, there is a single cycle of G containing both e and f
DUE: IN CLASS WEDNESDAY 2 JULY 2014.
1. Explicitly determine the elements of the cycle spaces of the following
(a) the complete graph K5 ;
(b) the complete bipartite graph K3,4 ;
(c) the 3-dimensional cube.
Rather than trying t
DUE: IN CLASS WEDNESDAY 18 JUNE 2014.
1. Recall that we dened a graph to be k-connected if: (i) |V (G)| k + 1
and, (ii) for any subset W of V (G) with |W | = k 1, G W is
connected. Let G be a graph and let k be a positive integer, at l
DUE: IN CLASS WEDNESDAY 4 JUNE 2014.
My apologies for mixing up the notation + (U ) on Assignment
3 and in the notes (p.27). From now on, I will use the notation of
the notes: for a subset U of vertices of a directed graph D, + (U )
DUE: IN CLASS WEDNESDAY 11 JUNE 2014.
1. Let k be a positive integer and let D be a directed graph. Suppose u, v,
and w are distinct vertices of D so that there are k pairwise arc-disjoint
directed uv-paths in D and there are k pairwis
DUE: IN CLASS WEDNESDAY 21 MAY 2014.
1. Let M be a matching in a graph G. Let T be a set of vertices in G so
|M | = |V (G)| odd(G T ) + |T | .
Prove that |M | = (G).
2. Let u be an unavoidable vertex in a graph G and let v be
DUE: IN CLASS WEDNESDAY 28 MAY 2014.
1. Let A denote the set cfw_v1 , v2 , . . . , vk of vectors in Rn . (Here k is just
telling you how many vectors there are.) The rank r(A) is the size of
a largest independent subset of cfw_v1 , v2
DUE: IN CLASS WEDNESDAY 14 MAY 2014.
1. Determine the number (G) of edges in the largest matchings in each
of the following graphs G. Also nd the avoidable and unavoidable
vertices in them.
(a) The complete graph Kn ;
(b) the complete