This document outlines the proof of a theorem in two cases: 1) The extremal case, where the graph G contains a subset A of vertices such that |A| is close to (1+a)n and the degree of A is less than a. 2) The non-extremal case, where the graph is shown to contain many complete k-partite subgraphs and (k-1)-paths are used to connect them, forming a (k-1)th power of a cycle that covers most vertices. Any remaining vertices I are added in a way that preserves this structure. The main tools used in the proof are described, including results on finding complete k-partite subgraphs

26. 2. Outline of the Proof
Our proof has two cases. One of them is the almost extremal case, the other
one is the far-from-extremal case. More precisely, we shall fix a constant
a > 0 and the two cases will be:
a-Extremal Condition (a-EC): A graph G is a-extremal, if there
exists an A c V(G) for which
~
• ct - a)n ~ IAI ~ (i +a)n and
• d(A) <a.
In this case we say that the set A and the graph G are a-extremal, otherwise
we say that they are a-non-extremal.
Non-Extremal Case: The basic fact that we make usc of in the non-
extremal case is that if we go around a complete k-partitc graph picking
vertices from each of the color classes cyclically, as one can see in Fig 1, then
we end up with a (k - 1)-path. We use this fact repeatedly throughout the
paper. In Section 4, using the tools developed in Section 3 , we first cover
a constant fraction of the vertices in G by Kk+I(t)'s and then we cover the
maximum number of the remaining vertices with Kk(t)'s where t = clogn
forra constant 0 < c < 1. We refer to the sets of Kk+1(t)'s a.nd Kk(t)'s by C
and K respectively.
We would inevitably be left with a set I of vertices that cannot be covered
in such a manner. However, we show that the number ofsuch vertices is smalL
Then, in Section 3.2, we prove a connecting lemma and use it to "connect"
the complete (k + 1)-partite graphs and the complete k-partite graphs by
(k - 1)-paths of length at most 9(k + 1)!. As noted above, after connecting
the graphs inC and K we get (k - 1)th power of a cycle covering most of the
vertices in V(C) U V(K) (see Figure 1).
While connecting the blocks,(small complete multipartite blocks), because
~the connecting (k- 1) path has at most 9(k +1)! vertices, and - though it
may use vertices from inside the blocks, that will influence at most 9(k +1)!
columns. The blocks may become unbalanced and we cut them to the right
size.
In the process, we move a small number of vertices from V(C) U V(K) to
I. To accommodate the vertices in I, using the procedure in Section 4.1.2,
we insert them in b.et,ween vertices of V(C) UV(K) so that the resulting cycle
is the (k - l)th power of a Hamiltonian cycle.
4
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Figure 1: Non-Extremal Case: Dashed lines represent the (k- 1)-paths constructed
using the Connecting Lemma.
Extremal Case: When our graph G is a-extremal, we use a relatively
simple Konig-Hall type argument to find the (k - l)th power of a Hamiltonian
cycle in G. We deal with the extremal case in Section~-
3. Main Tools
We shall assume that n is s~ciently large and use the following main pa-
rameters:
O~«a«l, (3)
where a « b means that a is sufficiently small compared to b. In order to
present the results transparently we do not compute the actual dependencies,
although it could be done.
3.1. Complete k-Partite Subgraphs
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Lemma 2 (Theorem 3.1 on page 328 in [1]). There is an absolute constant
5