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YUFEI ZHAO: The probabilistic
method in combinatorics

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is a powerful method
that can be used

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to demonstrate the
existence of some object

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by introducing randomness.

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In this video, we'll look at
a basic example illustrating

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an application of the
probabilistic method.

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We'll show this
theorem over here,

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which says that if G is
a graph with m edges,

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then G has a bipartite subgraph
with at least m over 2 edges.

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So let me remind you the
definition of a bipartite graph

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is one where I can partition the
vertices into two halves or two

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sets of vertices.

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And all the edges go from
one side to the other.

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So that's an example
of a bipartite graph.

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And what this theorem is saying
is that if we start with any

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graph G with some number
of edges-- in this case,

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five edges--

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I can find a pretty
large subgraph

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by keeping at least
half of the edges of G.

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For example, we can do this,
keeping four of the 5 edges

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to get this subgraph, G prime.

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And this graph, G
prime, is now bipartite

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because I can put these
two vertices on one

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side and these two
vertices on the other side,

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so that you see all the
edges go from the red

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or the pink vertices to the
green vertices, and none

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between the pink and
none between the greens.

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So this is an example of
finding a large bipartite

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subgraph in a graph G.

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Now, let me prove this
theorem, illustrating

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the probabilistic method.

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So we're given this graph
G. And what we'll do

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is assign a color, which will be
either black or white uniformly

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and independently at random
to each vertex of the graph G.

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So imagine flipping a
coin for each vertex

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and coloring each
vertex, Black or white.

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After doing this, we can put
all the white vertices of G

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on one side and all the black
vertices of G on the other side.

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OK.

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So now, G has some
number of vertices.

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And there will be some
edges going across

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from white to black, some edges
within white, and some edges

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within black.

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There could be a lot more edges.

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But that's an example of what
this graph might look like.

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And now, let me take G prime
to be the subgraph consisting

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of edges with white on one end
and black on the other end,

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so two different colors
on the two edges, two

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endpoints of this edge.

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This is a subgraph.

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It's a subset of the edges of
G. And note that every edge of G

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has probability 1/2 of being in
G prime because, for this edge,

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there are four different
possibilities for what the point

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colors might look like.

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And these four possibilities
are equally likely.

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And exactly two of
them will create

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an edge that falls in G prime.

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Now, knowing this by
linearity of expectations,

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the number of edges of G prime--

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here, G prime is
a random subgraph.

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Where G is given,
G prime depends

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on the coin flips that
we made to assign colors.

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So it's a random subgraph.

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So the number of edges of G
prime is a random variable.

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But this random variable
has an expectation.

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And we can compute this
expectation using linearity

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of expectations because each
edge falls in G prime with

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probability 1/2.

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And therefore, the
total expected number

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of edges in G prime is m over 2.

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This is the expectation.

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This is what happens on average.

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Thus, some instance of G prime
has at least m over 2 edges.

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If on average we
get m over 2 edges,

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then there must be some instance
with at least m over 2 edges.

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And this G prime is
what we're looking for.

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It has at least m over 2 edges.

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And it is bipartite
because we've

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kept all the edges from the
black side to the white side.

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And so the prime is bipartite.

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This finishes the
proof of the theorem.

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And you see that by
introducing randomness,

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we can prove something about
the existence of a structure

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with desired properties.

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And it's a hallmark of
the probabilistic method.

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And it's an important method
that has lots of applications.

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This is one of the simplest
examples of its applications.

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