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Line Graphs and Edge Coloring
Recap of previous lecture, we have discussed
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the elementary properties of Sub division
and Minor of a graph Kuratowski's Theorem,
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Wagner's Theorem and also proved Non-planarity
of Peterson Graph. Content of this lecture,
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we will discuss the Line Graphs and how the
line graphs are used in Edge-coloring and
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1-factorizations of a graph .
So, let us see the concept of Line Graph and
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we will use it for the Edge-coloring . So,
many questions about the vertices have analogies
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for the edges. Independent sets have no adjacent
vertices and matchings have no adjacent edges,
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so, analogs.
We have vertices and corresponding problem
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in the edges are also there with a different
name. If no vertices which are adjacent called
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independent sets; then for the matching no
edges are adjacent. Now, vertex coloring partitions
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the vertices into independent sets; we can
illustrate partition the edges also into the
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matchings. So, these problems we can relate
via the line graphs. Here, we repeat the definition
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emphasizing our return to the context in which
the graphs may be may have the multiple edges.
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We use line graphs because the edge graph
is not a proper term.
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Now, let us see the Line Graph. Line Graph
of a graph G is represented as L of G is a
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simple graph whose vertices are the edges
of that graph G with the edges of line graph
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is nothing but when the edges of G, they are
meeting at a common point. Take this particular
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example , if this is the graph G where we
have labeled the edges with their names. So,
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for every edge there will be a vertex in the
Line Graph. Now, as far as a edge in a line
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graph says that; whenever these 2 edges in
the original graph are meeting for example,
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d and e. So, they meet. So, they will form
an edge in the line graph.
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Similarly, d and f, they meet, so, d and f
as an edge. Similarly, e and f meets, so,
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e and f as an edge . Similarly, d and g they
meet, so, d and g have an edge and f and h
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they meet. So, f and h they have an edge.
So, h and g they meet. So, h and g they have
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an edge and f and g they also meet. So, f
and g they have an edge. So, if we are given
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a graph g, we can obtain the line graph of
that particular graph using this concept.
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So, again we can take another example. In
this example, we are given A graph. So, for
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a we have an vertex, B we have a vertex, for
E we have a vertex, for C we have a vertex
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and for D we have a vertex . Now, A and B
they are meeting ,
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A and B they are meeting, so, we place an
edge. A and E they are meeting, we place an
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edge. A and C they are meeting , we place
an edge, then B and D they are meeting, B
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and C they are meeting, so, we place an edge.
B and D they are meeting , we place an edge.
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Now C and D they are meeting, so, we place
an edge.
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Similarly, E and C they are meeting, so, we
place an edge. So, this is called a Line Graph
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of this particular graph.
Now, the Edge-coloring problem arises when
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the objects being scheduled are the pairs
of the underlying elements. So, example of
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edge colorings of let us say k 2n . So, k
2n when n is let us say 2. So, this becomes
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k 2n graph. Now, re-bond a pair of objects
to be considered as per the scheduling of
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a game is concerned, then it requires an edge
coloring not the vertex coloring. So, in the
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league of 2n teams, you want to schedule the
game, so that each pair of teams play a game
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, but each team plays at most once in a week.
So, since team may must play 2n minus 1, others
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the season last at 2n minus 1 week. So, the
games of each week must form a matching, we
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can schedule the season in 2n minus weeks
if and only if we can partition this edges
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into 2n minus 1 matching, since k 2n is 2n
minus 1 and regular, there must be a perfect
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matching. So, we can see here that having
arranged the bipartite graph instead of that
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let us add. So, this becomes k 2n graph. So,
k 2n graph if we take a particular team. So,
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how many because it is 2n minus 1 regular
graph . So, it can have a pairing with all
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n minus 1 vertices. So, these edges will pair.
Similarly, the other edges will also have
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such pairings. So, different pairings are
basically possible. So, we can partition these
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set of edges into 2n minus 1 different matchings
and how to obtain this matchings that is the
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pairings is called an Edge basically coloring.
So, when we obtain a pairing, so, there is
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a rule that if these 2 are selected then the
remaining pair; that means, this is one set
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of match when 1, 2, 3, 4 when 1 and 3 is one
such of matching and 2 and 4 is one matching.
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Second matching we can obtain as 1, 4 and
2, 3 is another matching.
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Similarly, we can have another matching like
1, 2 and 3, 4 n. So, this particular graph
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as 2n minus 3 different matchings and this
will require 2n minus 1 , so, 2n minus 1 matchings.
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So, here 2n minus 1 comes out to be 3. So,
3 matchings we have obtained. So, for that,
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you might have seen that the graph has to
be a 2n minus 1 regular and it has a perfect
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matching then only 2n minus 1 different matchings
are possible.
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Therefore, let us see the definition of k
edge coloring of a graph is the labeling of
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the edges with the color set S, where the
cardinality of axis k so; that means, applying
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k different colors, we are placing the colors
on these edges.
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And k edge coloring is nothing but labeling
or a function . So, the labels are the colors.
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The edges of one color will form the color
class, so k edge coloring is proper if the
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incidence edges have different labels, that
is if each color class is a matching. So,
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let us understand what is the proper coloring
the incident edges have different labels,
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so, take this particular graph. So, the incident
edge on this particular vertex if this is
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having label let us say red. So, the other
edge will have another label why because , the
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incident edges should have the different labels
for a proper . So, it is a proper 2 colorings
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2 edge colorings.
So, a graph is k-edge-colorable if it has
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a proper k edge colorings. The edge chromatic
that is , chi prime G of a loop less graph
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is the least value of k such that the graph
is k-edge-coloring. Example of edge-coloring
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a complete graph is basically shown over here.
Now, we call the edge chromatic number also
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as the chromatic index. So, chi prime G is
called chromatic index, since the edges sharing
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a vertex need different colors. Therefore,
this bound on this edge chromatic number that
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is chi prime G is at least the maximum degree
of a particular graph. Vizing and Gupta independently
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proved that delta G plus 1 colors suffice
when the graph is simple.
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So, we will see that what are the conditions
when the edge chromatic number of the graph
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that is the chi prime G is at least delta
and whatever the conditions as per as Vizing
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and Gupta Theorem that this particular chromatic
index will become delta G plus 1 . Now, a
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clique in the line graph in the set of pair
wise intersecting edges of G; because the
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vertices are representing the edges of graph
G. So, there is a clique in line graph; this
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means, there is a set of pair wise intersecting
edges of a graph. When G is simple, such edges
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form the star or a triangle in particular
G for the hereditary class of line graph of
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a simple graph Vizing Theorem states that
chi prime edge is less then or equal to omega
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h plus 1. Thus, the line graphs are almost
perfect .
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In contrast to chi G, multiple edges is greatly
affect the chi prime G that is chromatic index.
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So, graph with a loop as no proper edge-coloring.
So, that is important point to be noted.
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Hence, the loop less will be excluded here,
but multiple edges are allowed in the edge-coloring.
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So, multiplicity multiple edges means between
2 vertices multiple edges can be there and
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how many edges are there that is called multiplicity
. So, in a graph G with multiple edges, we
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say a vertex pair x, y is an edge of a multiplicity
m if there are m different edges with the
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end points x and y.
So, here let us say there are m. So, hence
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the multiplicity of m will be there . Now,
we write of mu of x y for multiplicity of
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a pair and we write down mu G for the maximum
of the edge multiplicities in the graph. So,
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this kind of structure where multiplicities
of where multiplicity of a graph or the multiplicity
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of the pair are allowed and this is called
a Fat Triangle.
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Now, there is a theorem which is given by
Konig, which states that if the graph G is
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bipartite, then chi prime G is equal to delta
G.
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So, let us see the proof corollary which is
stated in the previous videos, 3.1.13 states
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that, that is the Halls Theorem states that,
every regular bipartite graph H has one factor.
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So, it is a it is a Halls Theorem about matching.
So, one factor is nothing but a matching,
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so, every regular bipartite graph H as one
factor. So, by induction on big delta of H,
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this yields a proper delta H edge colorings
if therefore, suffices to show every bipartite
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graph G with a maximum degree k, there is
a regular k bipartite graph containing G.
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To construct such a graph, we add the vertices
to a smaller partite set of part G if necessary
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to equalize the sizes .
If the resulting graph G prime is not regular,
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then each partite set vertices less then that
degree k at that edge and thus make it s k-regular.
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For a regular graph G, the proper edge-coloring
with big delta G colors is equivalent to decomposition
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into one factor let us see through an example
. So, let us say that this is the graph which
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is given a bipartite. Now, here one vertex
is missing, we can add that vertex to a smaller
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partite set this is a smaller partite set,
we have added one vertex to equalize the sizes.
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Now, if this particular G prime is not regular,
then we will place an edge these edges are
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added extra and hence these becomes 2 regular
or let us say it is a k regular graph .
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Now, having obtained; that means, given any
such partite graph, we can obtain a k regular
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graph by these sort of operations by adding
vertices and by adding edges having done that?
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Now, you know that the big delta G of this
particular graph is 2. So, we can obtain an
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edge coloring with that most big delta G colors.
So, let us say that if these this edge in
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color, then we cannot color the other edge
which are incident on these vertices with
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the same color. So, the other 2 vertices which
need to be colored the other 2 vertices will
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require the same color, which are not same.
Now, the remaining edges require another color.
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So, hence we are using a green color. So,
the remaining edges we can apply the green
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color. So, every vertex which is incidence
having the edges of different colors. How
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many colors? That is big delta G the colors
will be required. Let us read again. So, corollary
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3.1.13 it states that, every regular bipartite
graph as one factor. So, that is basically
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the perfect matching if the graph is regular
bipartite graph. Now, by the induction on
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big delta of this particular graph this yields
proper delta edge edge-colorings. So, we have
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shown that and we have also shown through
these particular steps that, if the graph
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is not k regular, we can obtain it by adding
the vertices or adding the edges.
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So, hence this chronic, we have seen the theorem
that if graph G is bipartite, then chi prime
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G is equal to big delta G that is for bipartite
graphs. So, for bipartite graph, this particular
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bound holds according to the Chronic Theorem.
Let us see 1-factorization is nothing but
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the matchings the decomposition of a regular
graph into the 1-factor is the 1-factorizations
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of a graph G . So, graph with 1-factorization
is called 1-factorable. So, an odd cycle is
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not 1-factorable. So, take an odd cycle , this
is C 5 it is a odd cycle it is not 1-factorable;
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why because, if you can select this edge,
then this edge cannot be selected this edge
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if it is selected this vertex will be unsaturated.
Hence this is not the perfect matching. Hence,
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it is not 1-factorable. Similarly, if an even
cycle if we we can take, so we can factorize,
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we can obtain the 1-factor in the following
way . So, this is these are all called 1 factors
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of this particular graph 1, 2, 3, 4 . There
is another one factor, you can obtain between
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1, 3 and 2, 4 . So, this is 1 factorable.
So, the Petersen graph also requires an extra
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color, but only one extra color that is important
point to note. So, that means, the Petersen
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graph
is 3-regular, but it requires how many colors
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chi prime of Petersen is equal to 4; that
means, it requires big delta G is equal to
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3 plus 1 more. So, Petersen graph is is an
example where chi prime G becomes plus 1.
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So, Petersen graph is 4-edge-chromatic. So,
Petersen graph is 3-regular 3-edge-color ability
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requires 1-factorizations deleting a perfect
matching leaves 2 factors, all components
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are cycle. Here, we can see if you delete
it, this is a perfect matching. So, we will
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obtain 2 different cycles, which I will show
you through this is 1-cycles C 5 and this
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is another cycle. So, 2 cycles 2C 5 cycles
, we will obtain. All components cycles the
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1-factorizations can be completed only if
these are all even cycles. Thus, it suffices
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to show every 2-factor is isomorphic to 2C
5. So, here we have seen that consider the
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drawings of 2C 5 and a matching.
The cross edges between them we consider the
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cases by the number of cross edges are used.
So, every cycle uses an even number of cross
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edges. So, a 2-factor-edge has an even number
m of cross edges. If m is equal to 0, then
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H is equal to then H is equal to 2C 5 . Now,
when m is equal to 2 , so, here we can see
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this is 1 and 2 then the 2-cross-edges have
non-adjacent end points on the inner circle
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or the outer circle. On the cycle where their
end points are non-adjacent, the remaining
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3 vertices 4 or 5-edges of that cycle into
edge which violates the 2-factor requirements.
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So, when m is equal to 4 1, 2, 3, 4, then
the cycle edges forced into H by unused cross
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edge form a 2ps, this is one p and this is
another p whose only completion to 2 factor
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in H is 2cs. Note that, since c 5 has 3-edge-colorable
the graph is 4-edge-colorable.
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Vizing's theorem, Vizing and Gupta has given
a theorem which says that, if G is a simple
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graph, then chi prime G is less than or equal
to big delta G plus 1. Big delta G means,
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the max degree of particular graph is 1 . Now,
if we correlate with the previous discussion
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that we have seen that bipartite graph is
having chi prime G equal to big delta that
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also is included in to it. So, with this,
we include the definition; a simple graph
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G is class 1, if chi prime G is equal to big
delta. It is called class 2, if chi prime
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G is equal to big delta G plus 1. So, the
example here is a bipartite graph
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by Konig's theorem , we have seen this and
this becomes valid for a Petersen graph .
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Hence, the bipartite graph is class 1 graph
and Petersen's graph is class 2 graph. Determining
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whether a graph is class 1 or class 2 is generally
hard. Thus, we seek the condition that orbit
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or guarantees big delta G-edge-colorability.
So, conclusion; in this lecture, we have discussed
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the characterization of a Line Graphs, Edge-colorings,
Chromatic Index, Multiplicity and 1-factorizations.
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Thank you .