DIGITAL TEXT BOOK

1. 1
2015
MATCHINGS
NISHA RAJAN

2. 2
MATCHINGS

3. 3
MATCHINGS
NISHA RAJAN
DEPARTMENT OF MATHEMATICS
MOUNT TABOR TRAINING COLLEGE
PATHANAPURAM

4. 4
Contents
Preface
Introduction
1. Matchings
2. Independent and dominating sets
2.1. Independent sets and covers
2.2. Dominating sets
3. Matching in general graphs
Bibliography

5. 5
Preface
In the present times Mathematics occupies an important place
in curriculum. The “Matchings” is one of the important subject at MSc level.
In this book an attempt has been made to cover up most of topics included in
the MSc syllabus by Indian Universities. This book covers the syllabi of
Regular and Correspondence courses of Indian Universities.
The book has been written in a simple and lucid manner and is
up-to-date in its contents. To illustrate theory some examples have been
given.
It is hoped that the book will be appreciated by teachers and
students alike. While preparing the book, material has been drawn from
works of different authors, periodicals and journals and the author is indebted
to all such persons and their publishers.
All suggestions for improvement of the book shall be thankfully
accepted.
Author

6. 6
INTRODUCTION
Graph theory is an important mathematical tool in a wide variety
of subjects, ranging from operational research and chemistry to genetics
and linguistics and from electrical engineering and geography to
sociology and architecture. Graph theory has been independently
discovered many times through some puzzles that arose from the
physical world, consideration of chemical isomers, electrical networks
etc. One of beauties of graph theory is that it depends very little on
other branches of mathematics.
In the mathematical discipline of graph theory, a matching or
independent edge set in a graph is a set of edges without common
vertices. It may also an entire graph consisting of edges without common
vertices. There are number of graph parameters which arise as particular
cases of some general set- theoretic problem known as covering,
partitioning etc. Though the concepts to be discussed in this topic,
usually known as matchings, factors and factorization, lead to some such
parameters. The definitions and even many of results to be presented and
are applicable with very little change to multigraphs and even general
graphs.
In the first chapter we study about the definitions of matchings, its
example and Hall's Marriage theorem. second chapter concerns with
independent and dominating sets in matching. Finally the third chapter
concerns with matchings in general graphs and Tutte's theorem on 1-
factors.

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Chapter 1
MATCHINGS
Definition 1.0.1. A matching of graph G is a subgraph of G. Such that
every edge shares no vertex with any other edge. That is, each vertex in
matching M has degree one.
Definition 1.0.2. The size of a matching is the number of edges in
that matching.
Example 1.0.1. Consider the graph, we denote the edge that connects
vertices i and j as (i, j). Note that (3,8) is a matching. Obviously we can
get more. The pairs (3, 8), (4, 7) also make a matching. That is a
matching of size two. And a matching of size three is (2, 3), (4, 8), (5, 7).
A matching of size four means that every vertex is paired, but vertices 1 and
2 must both be paired with vertex 3. So three is the best matching we can
do.
Figure 1
2
1
8
7
6
5
4
3

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Definition 1.0.3. A matching is maximum when it has the largest
possible size. Note that for a given graph G,there may be several
maximum matchings. In figure 1, three is the maximum matching.
Definition 1.0.4. The matching number of a graph is the size of a
maximum matching of that graph.Thus the matching number of the
graph in figure 1 is three.
Definition 1.0.5. A matching of a graph G is complete if it
contains all of G's vertices. Sometimes this is also called a perfect
matching.
Example 1.0.2. perfect matchings in Kn,n . Consider Kn,n with partite
sets X=x1,x2,…….,xn and Y=y1,y2,……..,yn. A perfect matching defines
a bijection from X to Y. Successively finding mates for x1,x2,……., yields
n! perfect matchings.
Each matching is represented by a permutation of [n] , mapping i
to j when xi is matched to yi . We can express the matchings as matrix with
X and Y indexing the rows and columns , we let position i,j be 1 for each
edge xiyi in a matching M to obtain the corresponding matrix. There is one
1 in each row and each column.
X x1 x2 x3 x4
Y y1 y2 y3 y4 0 0 1 0
Figure 2 1 0 0 0
0 0 0 1
0 1 0 0

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Definition 1.0.6. An alternating path is a path that alternates
between matching and non-matching edges.
Definition 1.0.7. An augmenting path is an alternating path that
starts and ends on unmatched vertices.
Example 1.0.3. In the left half of a figure 3, we have gone down through
the entirety of L , but clearly an augmenting path exists between vertices L4
and R4. Now by breaking old matchings and adding new ones, we can
augment our matching into a matching that is larger by one. Basically
what we do is to reverse the state of all edges in our path, that is change
unmatching edges into matching ones and viceversa.
Left Right
Figure 3
Definition 1.0.8. If G and H are graphs with vertex set V , then the
symmetric difference GΔH is the graph with vertex set V whose edges
are all those appearing in exactly one of G and H.
L1
R4L4
R3L3
R2
R1
PO
UU
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
YY
L2
R4
R3
R2
R1
L4
L3
L2
L1

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Definition 1.0.9. The vertices incident to the edges of a matching M
are saturated by M, the others are unsaturated.
Lemma 1.0.1 . Every component of the symmetric difference of two
matchings is a path or an even cycle.
Proof. Let M and M' be matchings and let F = MΔM'. Since M and
M' are matchings, every vertex has atmost one incident edge from each of
them. Thus F has atmost two edges at each vertex. Since Δ(F) < 2, every
component of F is a path or a cycle. Further more , every path or cycle in F
alternates between edges of M — M' and edges of M' — M. Thus each cycle
has even length with an equal number of edges from M and from M'.
Theorem 1.0.1. A matching M in a graph G is a maximum matching in
G if and only if G has no M-augmenting path.
Proof. Let G has a matching larger than M if and only if G has an M-
augmenting path. We have observed that an M-augmenting path can be
used to produce a matching larger than M.
Conversely, let M' be a matching G larger than M; we construct an M-
augmenting path. Let F = MΔM'. By lemma 1.0.1, F consists of paths
and even cycles, the cycles have the same number of edges from M and
M'. Since |M| > |M'| , F must have a component with more edges of M'
than of M. Such a component can only be a path that starts and ends with
an edge o f M ' ; t h u s i t i s a n M - a u g m e n t i n g p a t h
i n G .
Definition 1.0.10. A vertex cover of a graph G is a set Q C V(G)
that contains atleast one end point of every edge. The vertices in Q
cover E(G).

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Definition 1.0.11. For any set S of vertices in G , the neighbour
set of S in G , denoted by N(S) is defined to be the set of all vertices
adjacent to vertices in S.
Theorem 1.0.2. (Hall's Marriage theorem). Let G be a bipartite graph
with partite sets X and Y. Then X can be matched into Y if and only if
|N(S)| ≥ |S| for each S C X.
Proof. First, suppose that X can be matched into Y and let S be a subset of X.
Since S itself is also matched into Y, we see immediately that |N(S)| ≤ |S|
Conversely, assume |S| < |N(S)| for each subset S C X and let M be a
maximum matching in G. Denote by XM and YM be the vertices of X and Y
that are covered by M. We claim that XM = X.
By contradiction, suppose X0 = X — XM is the set of vertices of X that are
not covered by M and let Y1 be the neighbourhood of Xo. If Y1
contains a vertex which is not covered by M, then obviously M is not a
maximum matching. Therefore, Y1 C YM and we may defined the set X1 of
vertices of XM that are matched with the vertices of Y1. Denoting i = 1, we
proceed according to the following procedure: as long as N(Xi) C YM be
increment i by one, define Yi = N(Xi-1 — (Y1 U Y2 U……U Yi-1) and
Xi be the set of vertices of XM, that are matched with the vertices of Y.
The procedure stops when we reach a set Xk that has a neighbour in Y
which is not covered by M. Notice that at each step of the procedure ,
|X0 U X1 U……..U Xi|>|X0 U X1 U……..U Xi-1| and hence the sets

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X1, X2,….. Xk C XM and Y1, Y2,…Yk CYM are non-empty. We now
define a sequence of vertices yk+1, xk, yk, xk-1, yk-1……… y1,x0
according to the following procedure:. let yk+i be a neighbour of Xk
which is not covered by M, x — k be a vertex of Xk adjacent to yk+1 and
i = k. as long as i > 0, define yi to be the vertex of Y which is matched
with xi, decrease i by one and let xi be a vertex of Xi which is adjacent
to yi+1.
In the sequence yk+1, xk, yk, xk-1, yk-1……… y1,x0 neither x0 nor yk+i are
covered by M and for each i = 1, 2, 3, k the edge xiyi belongs to M. But
then by replacing in M the edge xiyi by the edges xiyi+I(i = 0,1, 2,…. k).
We obtain a matching which is strictly greater than M. This
contradiction shows that XM = X.
Corollary 1.0.1. Let G be a k-regular bipartite graph with k > 0.
Then G has a perfect matching.
Proof. Let G have bipartition V = X U Y. There are |X| vertices in X
and each of these vertices has k edges incident with it. Thus there are
k x |X| edges going from X to Y. Similerly since each of the |Y| vertices
in Y has k edges incident with it there are k x |Y| edges going from Y
to X. By the bipartite nature of G each edges. Since k > 0 and k|X| =
k|Y|, cancelling k gives,
|X|= |Y|
Now, let S be a subset of X. Let E1 denote the set of edges incident
with vertices in S. Then, by the k-regularity of G,
|E1|= k|S| (1)

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Let E2 denote the set of edges incident with vertices in N(S). Then,
since N(S) is the set of vertices which are joined by edges to S, we have
El C E2. Thus
|E1| <|E2| (2)
Moreover by the k-regularity of G, we have
|E2|=k|N(S)| (3)
Using (1),(2) and (3), we get
k|N(S)|= |E2|>|E1|=k|S|
and so
k|N(S)| > k|S|
Since k > 0 this gives |N(S)| > |S|. Since S was an arbitarary subset of
X, it follows from Hall's Marriage theorem that G contains a matching M
which saturates every vertex in X. Since |X| = |Y| the edges in the
matching M also saturate every vertex in Y. Thus M is a perfect
matching in G.

14. 14
Chapter 2
INDEPENDENT AND
DOMINATING SETS
2.1 Independent sets and covers
Definition 2.1.1. A set of edges in a graph is independent if no two
edges in the set are adjacent.
Definition 2.1.2. The vertex independence number β(G) of a
graph G is the maximum size of an independent set of vertices in G.
Definition 2.1.3. The edge independence number β' (G) of a
graph G is the maximum size of an independent set of edges.
Example 2.1.1. The independence number of a bipartite graph doesnot
always equal the size of partite set. In the graph below, both partite sets have
size three, but we have marked an independent set of size four.
Figure 1

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Definition 2.1.4. A vertex cover in a graph G is a set of vertices that
covers all edges of G.
Definition 2.1.5. A vertex covering number α(G) of a graph G is the
minimum number of vertices in a vertex cover of G
.Definition 2.1.6. An edge cover of G is a set L of edges such that
every vertex of G is incident to some edge of L.
Definition 2.1.7. The edge covering number α’(G) of a graph G is the
minimum size of an edge cover of G.
Example 2.1.2. In the graph on left below we mark a vertex cover of size
two and show a matching of size two in bold. The vertex cover of size two
prohibits matchings with more than two edges and the matching of size two
prohibits vertex covers with fewer than two vertices. As illustrated on the
right, the optimal values differ by one for an odd cycle. The difference
can be arbitararily large.
Figure 2
Example 2.1.3. The graph below has 13 vertices. A matching of size four
appears in bold, and adding the solid edges yields an edge cover of size 9. The
edge cover consists of four stars: for each we extract one edge (bold) to form
the matching.

16. 16
Figure 3
Definition 2.1.8. A maximal matching in a graph is a matching that
Cannot be enlarged adding an edge.
Theorem 2.1.1. If G is a bipartite graph, then the maximum size of a
matching in G equals the minimum size of a vertex cover of G.
Proof. Let G be a X — Y bigraph. Since distinct vertices must be used
to cover the edges of a matching, |Q| > |M|. Whenever Q is a vertex. The
minimality of Q thus yields Hall's condition in H and hence H has a
matching that saturates R. Applyng the same argument to H’ yields the
matching that saturates T.
Theorem 3.1.2. For every graph G of order n containing no isolated ver-
tices, α'(G) + β'(G) = n.
Proof. First we suppose that β’(G) = k. Then a maximum matching of G
consists of k edges, which then cover 2k vertices. The remaining n — 2k
vertices of G can be covered by n — 2k edges. Thus
α’(G)<k+ (n — 2k) = n — k.
Hence,
α’(G)=β’(G)< (n — k)+ k = n
It remains only to show that α'(G) + β'(G) > n. Let X be a minimum
edge cover of G. Hence |X| =l= α'(G). Consider the subgraph F = (X)

17. 17
induced by X. We begin with an observation : cover and M is a
matching in G. Given a smallest vertex cover Q of G, we construct a
matching of size |Q|, to prove that equality can always be achieved.
Partition Q by letting R = Q ᴒ X and T = Q ᴒ Y. Let H and H' be the
subgraphs of G induced by R U (Y — T) and T U (X — R) respectively.
We use Hall's theorem to show that H has a matching that saturates R
into Y-T and H' has a matching that saturates T. Since H and H' are
disjoint, the two matchings together form a matching of size |Q| in G.
Since RUT is a vertex cover, G has no edge from Y—T to X — R. For
each S C R, we consider NH(S) which is contained in Y — T.If
|NH(S)|<|s|, then we can substitute NH(S) for S in Q, to obtain a smaller
vertex cover. Since NH(S) covers all edges incident to S that are not
covered by T.
F contains no trail T of length three . If F did contain a trail T,
then X — {e} also covers all vertices of G, which is impossible.
Therefore, F contains no cycles and no paths of length three or more,
implying that every component of F is a star.
Since a forest of order n and size n — k contains k components and the
size of F is l = n — (n — l), it follows that F contains n —l non-trivial
components. Selecting one edge from each of these n — l components
produce a matching of size n — l, that is β'(G) > n — l. Therefore,
α’(G) + β' (G)≥ l+(n— l) = n
That is,
a' (G) +β '(G) = n

18. 18
Corollary 2.1.1. For every graph G of order n containing no isolated ver-
tices, then
α(G)+ β(G) = n
Definition 2.1.9. An independent set in G of size /3(G) is called a
maximum independent set.
Definition 2.1.10. A vertex cover of G of size a(G) is called a minimum
vertex cover in G.
Definition 2.1.11. An edge cover of G of size a' (G) is called a minimum
edge cover of G.
Example 2.1.4. Determine the values of α(G), β(G), α’(G) and β’(G) for
the graph G =̃ K1 + 2K3 of the following figure
u y
w
G:
t z
Figure 4
Solution: Since the order of G is 7, it follows that β'(G) = 3. Because
{tu, vw, yz} is an independent set of three edges, β'(G) = 3. By theorem
2.1.2, α’(G) = 4. Note that {tu, vw, wy, yz} is a minimum edge cover for G.
v x

19. 19
The vertex w is adjacent to all other vertices of G. Further more,
({t,u,v})=̃K3 and ({x, y, z}) =̃ K3. Thus β(G) = 2. One example of maximum
independent set is {t, z}. By corollary 2.1.1, α(G) = 5. One example of
a minimum vertex cover of G is {t, u, w, y,z}.
2.2 Dominating sets
Definition 2.2.1. In a graph G, a set S C V(G) is a dominating set if
every vertex not in S has a neighbour in S.
Definition 2.2.2. The domination number γ(G) is the minimum size
of a dominating set in G.
Example 2.2.1. The graph G below has a minimum dominating set of size
(circles) and a minimum dominating set of size 3 (squares): γ(G) = 3.
Example 2.2.2. Covering the vertex set with stars may not require as many
stars as covering the edge set. When a graph G has no isolated vertices,
every vertex cover is a dominating set, so γ(C) < β(G). The difference can be
large. γ(Kn) = 1, but β(Kn) = n —1.
Definition 2.2.3. A dominating set S in G is a connected
dominating set if G(S) is connected.
Definition 2.2.4. A dominating set S in G is an independent
dominating set if G(S) is independent.

20. 20
Definition 2.2.5. A dominating set S in G is a total dominating set
if G(S) has no isolated vertex.
Lemma 2.2.1. A set of vertices in a graph is an independent dominating
set if and only if it is a maximal independent set.
Proof. Among independent set, S is a maximal if and only if every
vertex outside S has a neighbour in S, which is the condition for S to be a
dominating set. ❑
Theorem 2.2.1. Every claw-free graph has an independent dominating
set of size γ(G).
Proof. Let S be a minimum dominating set in a claw-free graph G and
S' be a maximal independent subset of S. Let T = V (G) — N(S') and T'
be a maximal independent subset of S. Since T' contains no neighbour in
S', S'UT' is independent. Since S' is maximal in S, we have S C N(S').
Since T' is maximal in T, T' dominates T. Hence S' UT' is a dominating
set. It remains to show that |S' U T'| < γ(G). Since S' is maximal in S, T'
is independent and G is claw-free, each vertex of T' has atleast one
neighbour in S — S'. H e n c e | T ' | < | S — S ’ | w h i c h y i e l d s | S '
U T ’ | < | S | = γ ( G ) .

21. 21
Chapter 3
MATCHING IN GENERAL
GRAPHS
Definition 3.0.1. A k-factor of G is a k-regular spanning subgraph of
G. A 1-factor of G is a 1-regular spanning subgraph of G, which is also
called perfect matching.
Definition 3.0.2. An odd component of a graph is a component of odd
order; the number of odd components of H is o(H).
Definition 3.0.3. A graph G contains a 1-factor and a set S C V (G), then
every odd component of G — S has a vertex matched to something
outside it. That is, o(G — S) < |S| for all S C V (G) is called Tutte's
condition
Theorem 3.0.2. (Tutte'.s 1-factor theorem). A graph G contains a 1-
factor if and only if o(G — S)<|S| for every S C V(G).
Proof. Assume, first that G contains a 1-factor F. Let S be proper subset of
V(G). If G — S has no odd components , then o(G — S) = 0 and certainly
o(G-S) <|S| . Suppose that o(G— S) = k > 1 and let G1,G2,……Gk be
the odd components of G — S. Since G contains the 1-factor F and the
order of each subgraph Gi(1 < i < k) is odd, some edge of F must be
incident to both a vertex of Gi and a vertex S and so o(G — S) < |S|.
Conversely, assume that o(G — S) < |S| for every proper subset S
of V(G). In particular, for S =ф , we have o(G — S) = o(G) = 0, that is,
every component of G is even and so G has even order. We now show by

22. 22
induction that every graph G has even order with this property has a 1-
factor.There is only one graph of order two having only even components,
namely1c2,which has a 1-factor. Let G be a graph of order n satisfying
o(G — S) < |S| for every proper subset S of V(G). Thus every component
of G has even order. Since every non-trivial component of G contains a
vertex that is not a cut-vertex, there are subsets R of V(G) for which o(G-R)
= |R|. Let S be ofmaximum size and let G1, G2, ….Gk be the k odd
components of G — S.
Thus k =|S| > 1. Observe that G1, G2, ….,Gk are the only
components
of G — S, for otherwise, G — S has an even component Go containing a vertex
uo that is not a cut-vertex. Then for the set So = S U {uo} of size k+1,
o(G - So) =|S0|= k + 1
which is impossible. So as we claimed, the odd component G1, G2,….Gk
are the only components of G — S. Now, for each integer i with
1<i<k let S, be the set of vertices of S that are adjacent to atleast one
vertex in Gi. Since G has only even components each set Si is non-
empty. We claim next that for each integer 1 with 1 < l < k, the union
of any l of the sets S1, S2, , Sk contains atleast l vertices.
Assume the contrary that there exist an integer j such that the
union T of j of the sets S1, S2 ,…… Sk as fewer than j elements.
Without loss of generality, we assume that
T =S1 U S2 U ………. U Sj and |T| < j.
Then o(G — T) > j >|T|, which is impossible. Thus, as claimed for
each integer l with 1 < l < k, the union of any l of the sets S1, S2 ,……
Sk contains atleast l vertices. We know the result, a collection {S1,
S2,……….Sn}of non-empty finite sets has a system of distinct
representatives if and only if for each integer k with 1 < k < n, the
union of any k of the sets contains atleast k elements.

23. 23
Then we have, there exist a set V1, V2,…………Vk of k distinct
vertices such that Vi Є Si, for 1 < i < k. Since every graph Gi(1 < 1
< k) contains a vertex u, for which ui,vi, E Є (G), it follows that { ui,vi :
1 < i < k} is a matching of G.
Next we show that if Gi(1 < i < k) is non-trivial, then G, — u, has
a 1-factor. Let W be a proper subset of V(Gi — ui). We claim that
o (Gi — ui — W) ≤ |W|.
Assume the contrary that o(Gi — ui — W) > |W|. Since Gi — ui has
even order, o(Gi — ui — W) and |W| are either both even or both odd.
Hence o(Gi — ui — W) >|W| + 2. Let S' =SUWU{ui }. Then,
|S”
|≥o(G — S') = o(G — S)+ o(Gi — ui-W)-1
>|S|+ (|W|+ 2)-1 =|S|+ |W| + 1 = |S'|,
which implies that o(G — S') = |S'|, contradicting our choice of S. Therefore,
o(Gi — ui — W) < |W|.
By the induction hypothesis , if G,(1 < i < k) is non-trivial, then Gi — ui
has
a 1- factor. The collection of 1-factors of Gi — ui far all non-trivial
graphs
G (1 < i < k) and the edges in { ui,vi : 1 < i < k} produce a 1-factor of
G.
Definition 3.0.4. The Petersen graph is the simple graph whose
vertices are the 2-element subsets of a 5-element set and whose edges are the
pairs of disjoint 2-element subsets.

24. 24
Corollary 3.0.1. (Petersen’s theorem). Every 3-regular graph with no cut-
edge has a 1-factor.
Proof. Let G be a 3-regular graph with no cut-edge and let S be a subset of
V(G) of size k > 1. We show that the number o(G — S) of odd
components of G — S is atmost |S|. Since this is certainly the case if
G — S has no odd components. We may assume that G — S has l > 1
odd components G1,G2,……Gi. Let X1(1 < i < 1) denote the set of
edges joining the vertices of S and the vertices of Gi. Since every vertex
of each graph Gi has degree 3 in G and the sum of the degrees of the
vertices in the graph Gi is even, |Xi| is odd. Because G has no cut-edge,
|Xi| ≠ 1 for each i, 1 ≤ i ≤ l and so |Xi| ≥ 3. Therefore, there are atleast
3 l edges joining the vertices of S and the vertices of G — S. However,
since |S| = k and every vertex of S has degree 3 in G, atmost 3k edges join
the vertices of S and the vertices of G — S. Therefore,
3o(G-S) = 3l ≤ 3k = 3|S|
and so o(G — S) ≤ |S|. By Tutte's 1-factor theorem, G has a 1-factor.
Example 3.0.3. Construction of a 2-factor: Consider the Eulerian
circuit in G = k5 that successively visits 1231425435. The corresponding
bipartite graph H is on the right. For the 1-factor whose (u,w)-pairs are
12,43,25,31, the resulting 2-factor is the cycle (1,2,5,4,3). The remaining
edges form another 1-factor, which corresponds to the 2-factor (1,4,2,3,5)
that remains in G.

25. 25
1
5 2
4 3
Definition 3.0.5. A graph G is said to be 1-factorable, if there exist
1- factors F1, F2, …… , Fr of G such that E(F1), E(F2),……, E(Fr)
is a partition of E(G).
Theorem 3.0.3. The Petersen graph is not 1-factorable.
Proof. Assume the contrary that the Petersen graph PG is 1-factorable.
Thus PG can be factored into three 1-factors F1, F2, F3. Hence the
spanning sub-graph H of PG with E(G) = E(F1) U E(F2) is 2-regular and
so H is either a single cycle or a union of two or more cycles. On the
otherhand, since the length of a smallest cycle in PG is 5, it follows that
H ≅ 2C5. This is i m p o s s i b l e , s i n c e 2 C 5 d o e s n o t c o n t a i n a
1 - f a c t o r .
Theorem 3.0.4. Every r-regular bipartite graph, r ≥ 1 is 1-factorable.
Proof. Let G be an r-regular bipartite graph, where r≥1. We know that
every r-regular bipartite graph (r ≥ 1) has a perfect matching. Therefore G
contains a perfect matching M1.Hence G — M1 is (r — 1)-regular. If r ≥ 2,
then G — M1 contains a perfect matching M2. We see that E(G) can be
partitioned into perfect matchings, which gives rise to a 1-factorization
of G.

26. 26
Definition 3.0.6. A graph G is said to be 2-factorable if there exist 2-
factors Fl, F2, Fk such that {E(F1 ), E(F2 ), , E(Fk )} is a parti-
tion of E(G).
Theorem 3.0.5. A graph G is 2-factorable if and only if G is r-regular
for some positive even integer r.
Proof. We know that every 2-factorable graph is r-regular for some
positive even integer r. Therefore we have to prove the converse part.
Let G be an r-regular graph, where r = 2k and k ≥ 1. Assume that G is
connected. We have, a non-trivial connected graph G is Eulerian if and
only if every vertex G has even degree. Then G is Eulerian and therefore
contains an Eulerian circuit C.
Let V (G) = {v1, v2,….. vn}. We construct a bipartite graph H with
partite sets, U = {u1,u2,…..un} and W = {w1, w2, ….win}, where
the vertices ui and wj(1 ≤ i, j≤ n) are adjacent in H, if vj immediately
follows vi on C. Since every vertex of G appears exactly k times in C,
the graph H is k-regular. By theorem 4.0.4, H is 1-factorable and H can
be factored into k 1-factors F1’ F2’,…., Fk’.
Next we show each 1-factor F1(1≤i ≤ k) of H corresponds to a 2-
factor F, of G. Consider the 1-factor F1, for example. Since Fl is a perfect
matching of H, it follows that E(F1’) is an independent set of k edges of
H, say E(F1 ’)={ u1 wi 1 , u2 wi2 , …. . , un win}, where the integers il,
i2 ,…..,in are the integers 1,2....n in some order and ij≠j for each j ( 1 ≤ j
≤ n ). Suppose that it = 1. Then the 1-factor Fl’ gives rise to a cycle C(1)
:
v1,vi1,……vt,vit = v1. If C(1)
has length n , then Hamiltonian cycle CO)
is less than n, then there is a vertex vi of H that is not on C(1)
. Suppose
that is = l. This gives rise to a second cycle C(2)
: v1,vi1,……vs,vis = vi.
Continuing in this manner, we obtain a collection of pairwise vertex-disjoint
cycles that contain each vertex of H once, producing a 2-factor F1 of G. In
general, then the 1-factorization of H into 1-factors Fl’, F2’,….. , Fk’
produces a 2-factorization of H into 2-factors F1, F2,……Fk as desired.

27. 27
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[3] John Clark and Derek Allan Holton, A first look at Graph Theory, World scientific,
Newzeland, 1991.
[4] K R Parthasarathy, Basic Graph Theory, Tata McGraw- Hill, New York, 1994.
[5] G Suresh Sigh, Graph Theory, PHI Learning Private limited, New Delhi, 2010.