CONSTRUCTION OF CHROMATIC POLYNOMIALS ON TOTAL GRAP HS

R.V.N.SRINIVASARAO 1 , J.VENKATESWARARAO 2 & D.V.S.R.ANILKUMAR 3 1Department of Mathematics, Guntur Engineering College, Guntur Dt, A.P, India

2Department of Mathematics, Mekelle University Main Campus, Mekelle, Ethiopia 3Department of Mathematics, Nizam Institute of Engineering and Technology, Nalgonda (Dt), A.P, India

ABSTRACT

This manuscript determines the characterizations of Chromatic Polynomials of diverse graphs

using deletion contraction algorithm. It also widens the concepts of Chromatic Polynomial of Cycle

Graph of order n. Further, it establishes that the construction of Chromatic Polynomial on Total Graph T

(G) of a (p, q)-connected graph G with p 3, 1 q 3. Lastly some basic elementary characterizations of

Chromatic Polynomials on Total Graphs were established.

KEYWORDS : Chromatic Polynomial, Total Graph, Colouring, Complete Graph, AMS subject

classification: 05C15, 05C31, 05CXX.

INTRODUCTION

Coloring of vertices of a graph is a common problem in the study of graph theory. We colored

the vertices such that adjacent vertices have different colours, is called ‘proper colouring’. During the

period that the Four Color Problem was unsolved, which spanned more than a century, many approaches

were introduced with the hopes that they would lead to a solution of this prominent problem.

Frequently, we are concerned with determining the least number of colours with which we can

achieve a proper colouring on a graph. Also, we want to count the possible number of different proper

colourings on a graph with a given number of colours. We can calculate each of these values by using a

special function that is associated with each graph, called the Chromatic Polynomial introduced by

George David Birkhoff [1912].

Afterward Whitney [1932] expanded the study of Chromatic Polynomials from maps to graphs.

While Whitney obtained a number of results on chromatic polynomials of graphs. This did not contribute

to a proof of the Four Color Conjecture. R C. Read [1968] wrote a survey paper on chromatic

polynomials improved interest in chromatic polynomials of graphs. Also Meredith [1972] discussed

about the coefficients of chromatic polynomials.

Eventually P.Erdos and R.J.Wilson, (1977) studied on the chromatic index graphs. Latter

N.Biggs (1994) willful on algebraic graph theory. Further R.A.Brualdi (1999) discussed about the

deletion-contraction algorithm to find the Chromatic Polynomials. Newly F.M.Dong (2005) contributed

International Journal of Mathematics and Computer Applications Research (IJMCAR) ISSN 2249-6955 Vol.2, Issue 3 Sep 2012 92-105 © TJPRC Pvt. Ltd.,

93 Construction of Chromatic Polynomials on Total Graphs

to the Chromaticity of Graphs in his study. Recently, J.V.Rao and R.V.N.S.Rao [2012] deliberated on the

Upper Bounds of Chromatic Number of Total Graphs.

In the first section, we were cognizant on the basic definitions and established the Chromatic

Polynomials by the method of structural analysis with appropriate examples. We also discussed the

Brualadi (1999) deletion –contraction algorithm.

In the second section, we discussed the varied methods to find the Chromatic Polynomials.

Particularly we provided the proof for reduction theorem. We also extended the concept to find the

Chromatic Polynomial of Cycle Graph Cn.

In the third section, we extended the concept of Chromatic Polynomials to Total Graphs and

established the Chromatic Polynomials of Total Graphs of a Graph G of order at most three by using

deletion –contraction algorithm and its equivalent forms.

Finally, we discussed some basic algebraic characterizations of Chromatic Polynomials and

briefly explained about the application of Chromatic Polynomial to the scheduling problems.

PRELIMINARIES

All the graphs we considered are finite. Here we follow the notations of West D.B. Introduction

to the graph theory [2003].The terminology and concepts not presented here can be followed from. West

D.B.

Definition: [11] A Null graph is one in which the edge family, E (G) is empty. A null graph on n

vertices is denoted by Nn. See fig1

Fig 1 .A Null Graph on 3 Vertices

1.2. Definition:[11] A complete graph is a simple graph in which each pair of distinct vertices

are adjacent .A complete graph on n vertices denoted by Kn. See figure2

Figure 2: K4:A Complete Graph on 4 Vertices

Fig3: C4: A cycle graph on 4 vertices.

R.V.N.Srinivasarao,J.Venkateswararao & D.V.S.R.Anilkumar 94

Fig 4:P4:A path graph on 4 vertices

1.3 Definition:[11] A connected graph in which the degree of each vertex is 2 is a cycle graph

.A cycle graph on n vertices is denoted by Cn (see figure3).

1.4: Definition: [11] A path graph on n vertices is the graph obtained when an edge is removed

from the cycle graph Cn. A path graph on n vertices is denoted by Pn (see figure4).

1.5. Definition: [11] Let G be a graph, the contraction of a graph G with edge e connected by

vertices u and v is obtained by merging any two adjacent vertices u and v in to a single vertex and by

joining this new combined vertex to all those vertices to which either u or v we already adjacent, denoted

by G/e.(see figure5)

V1 v1v2

1.6 Example:

V2 v3

G G/e

Figure 5: Contracting the edge e from the graph G that is G/e

1.7. Definition: [11] A coloring of a graph G so that adjacent vertices are different colours is

called a proper colouring of the graph.

1.8. Definition: [11] A graph G is K-colorable if we can assign one of k colours to each vertex

to achieve a proper colouring.

1.9. Definition: [11] A graph G is K –chromatic if G is K- colorable but not (k-1) colourable,

the chromatic number of G is denoted by )(Gχ .

1.10 Remark. Now we discuss the chromatic polynomial of the graph G .This is special function

that describes the number of ways we can achieve a proper coloring on a graph G with given λ colours.

1.11. Definition:[11] For a graph G and a positive integerλ , the number of different proper

λ -colorings of G is denoted by P (G,λ ) and is called the chromatic polynomial of G. Twoλ -colorings

c and c' of G from the same set {I, 2, ... , λ } of λ colors are considered different if c(v) ≠ c'(v) for some

vertex v of G .It is also denoted as PG(λ ).Obviously if )(Gχ > λ then P (G,λ )=0.

1.12.Example.If we want to colour the null graph N3 with λ colours .we notice that this can be

done λ 3 ways since no vertex adjacent to another hence there are λ colour option for each vertex. In

general, we know that P (Nn,λ )= P(Nn,λ )= λ n. See the fig 6

e

95 Construction of Chromatic Polynomials on Total Graphs

λ

λ λ

Figure.6: Calculating Chromatic polynomial of N3

REMARKS

In general it is very difficult to determined the chromatic polynomial by analysis of the structure

of the graphs, as is done above .However, we provide a method for computing these functions by

deleting an edge in the graph and contracting the vertices connected by this edge .When we contract two

vertices, we identify them as a single vertex and all edges incident with either vertex become incident

with both.

Remark

Brualadi (1999) impart a formal algorithm known as deletion contraction algorithm for finding

a chromatic polynomial. The following figure 7 explain the deletion contraction algorithm with a graph

on four vertices

Figure7: Reducing a graph to its null graphs using Deletion and Contraction Algorithm

In figure 7, each step to the left represents a deletion, and the step on the right represents a

contraction. After each step, if the graph is not reduced down to a null graph, the algorithm is repeated

R.V.N.Srinivasarao,J.Venkateswararao & D.V.S.R.Anilkumar 96

.At the end of the algorithm, only null graphs remains. Since the chromatic polynomial of a null graph of

order n is n, we get the chromatic polynomial P (G, ) for the graph in figure 7 is 4-4 3-5 2-2 .

Diverse Methods to Find Chromatic Polynomials

In this section, first we have to prove the Deletion- Contraction algorithm which is also known

as reduction theorem as in theorem 2.1.

2.1. Theorem: Let G be a graph, and G-e and G/e, be the graphs obtained from G by deleting

and contracting an edge e respectively .Then P (G, ) =P (G-e, )- P(G/e, ).

Proof: Let e be an edge incident on the vertices uv. The number of -colourings of G-e in which

u and v have different colours is the same with or without edge e and is thus equal to P (G, ).

Similarly , the number of k-colourings of G-e in which u and v are the same colour does not change

regardless of whether the two vertices are contracted ,this number is thus equal to P(G/e, ). Also we

observe that the graph G/e may not be a simple graph ,but u and v are distinct vertices ,the contraction

will not create any loops. Also we can ignore multiple edges between vertices as this does not affect the

calculation of the chromatic polynomial. As a result we find the total number of -colourings of G-e is

P(G, )+ P(G/e, ), that is P(G-e, ) = P(G, ) + P(G/e, ), hence P(G, ) =P (G-e, ) - P(G/e, ). We utilize

figure 8 as reference to this theorem.

Figure 8: Chromatic polynomial of K4 by Deletion –contraction algorithm

-

97 Construction of Chromatic Polynomials on Total Graphs

EXAMPLE

The figure8 establish different stages to find the chromatic polynomial of Complete graph on 4

vertices, K4 using Deletion –Contraction algorithm .By applying repeatedly the theorem we reduce the

graph K4 to combination of path graph on 4 vertices, cycle graph on 3 vertices and path graph on 2

vertices .As we know the chromatic polynomials of the above said graphs, the chromatic polynomial of

K4 is

P (K4,λ ) =λ ( λ -1)3-λ ( λ -1)(λ -2)- λ ( λ -1)2- λ ( λ -1)(λ -2)

= λ 4-6λ 3 -11λ 2-6λ .

Since from the analysis of structure of graphs, that chromatic polynomial of K4 is P (K4, )

= 4-6 3 – 11 2-6 .Hence the chromatic polynomial for K4 by the method of analysis of the structure of

the graphs and by the method of reduction of graphs is same. But the reduction method is easy to find the

chromatic polynomial of any graph compare to the method of analysis of the structure of the graphs.

Lemma: The Chromatic Polynomial of a cycle graph of order n is (λ -1)n+(-1)n(λ -1). PROOF

This lemma can be proved by using mathematical induction on n. First we have to consider the

case n=3.the chromatic graph obtained by deleting an edge is ( -1)2.The chromatic polynomial of the

graph obtained by contacting the edge is ( -1).so by the reduction theorem the chromatic polynomial of

C3 is ( -1)2- ( -1)= ( -1)3-+(-1)3 ( -1).Hence the lemma is true for n=3.Suppose it is hold for all cycle

graphs of order n-1.Let G be any cycle graph of order n, and let e be one of its edges .Now the graph G-e

is a path with n vertices ,so its chromatic number is ( -1)n-1. The graph obtained from G by

contracting this edge is a cycle graph of order n-1 ,and its chromatic polynomial is ( -1)n-1 -+(-1)n-1 ( -

1) by induction hypothesis. So by applying the reduction theorem, the chromatic polynomial of G is

[ ( -1)n-1]-[ ( -1)n-1 -+(-1)n-1 ( -1)]= ( -1)n+(-1)n( -1).Hence the lemma is true for n as well.

2.4. Example: We now determine the chromatic polynomial of C4 in Figure 3. There are

choices for the color of v1. The vertices v2 and v4 must be assigned colors different from the assigned to

v1.The vertices v2 and v4 may be assigned the same color or may be assigned different colors. If v2 and

v4 are assigned the same color, then there are - 1 choice for that color. The vertex v3 can then be

assigned any color except the color assigned to v2 and v4. Hence the number of distinct -colorings of C4

in which v2 and v4 are colored the same is ( -1)2.If, on the other hand, v2 and v4 are colored

differently, then there are - 1 choice for v2 and - 2 choices for v4. Since v3 can be assigned any color

except the two colors assigned to v2 and v4, the number of -colorings of C4 in which v2 and v4 are

colored differently is ( - 1)( - 2)2. Hence the number of distinct -colorings of C4 is

R.V.N.Srinivasarao,J.Venkateswararao & D.V.S.R.Anilkumar 98

P (C4,λ ) = λ (λ - 1)2+λ (λ -1)( λ -2)2

= λ ( λ -1)( λ 2-3λ +3)

= λ 4-4λ 3+6λ 2-3λ

= (λ -1)4+(λ -1) REMARK

The earlier example 2.2 illustrates an important observation. Suppose that u and v are

nonadjacent vertices in a graph G. The number of -colorings of G equals the number of -colorings of

G in which u and v are colored differently plus the number of -colorings of G in which u and v are

colored the same. Since the number of -colorings of G in which u and v are colored differently is the

number of -colorings of G + uv while the number of -colorings of G in which u and v are colored the

same is the number of -colorings of the graph H obtained by contracting u and v, it follows that P(G, ) =

P(G + uv, ) + P(H, ).

THEOREM

Let G be a graph containing nonadjacent vertices u and v and let H be the graph obtained from

G by contracting u and v. Then P(G, ) = P(G + uv, ) + P(H, )

PROOF

Note that if G is a graph of order n ≥ 2 and size m ≥ 1, then G + uv has order n and size m + 1

while H has order n - 1 and size at most m. The equation stated in Theorem 2.1 can also be expressed as

P(G + uv, ) = P(G, ) -P(H, ).In this context, Theorem 2.1 can be rephrased in terms of an edge deletion

and an elementary contraction Hence P(G, ) = P(G + uv, ) + P(H, ).

CHROMATIC POLYNOMIALS OF TOTAL GRAPHS

In this section we have to locate Chromatic Polynomials of Total Graphs of a (p, q)-connected

graph G with p 3, 1 q 3. First we, define total graph of a graph G.

DEFINITION

Let G be a graph with vertex set V(G) and edge set E(G). The total graph of G, denoted by

T(G) is defined in the following way. The vertex set of T(G) is V(G) union E(G). Two vertices x, y in

the vertex set of T(G) are adjacent in T(G) if one of the following holds: (i). x, y are in V(G) and x is

adjacent to y in G.(ii). x, y are in E(G) and x, y are incident in G.(iii). x is in V(G), y is in E(G), and x, y

are incident in G.

To get the chromatic polynomial of total graphs of a (p,q)-connected graph with p ,q 3,consider

the entire possible total graph with these restriction and established chromatic polynomials in each case.

99 Construction of Chromatic Polynomials on Total Graphs

LEMMA

The chromatic polynomial of total graph of a (p,q)-connected graph G is = ( -1)[( -2)p+q-2+ (

-2)p+q-4] if p + q 4 and ( -1)[( -2)p+q-2] if p + q < 4, where p 3, 1 q 3.

Proof: The proof of this lemma can be come apart in to two cases with p + q < 4 and p + q 4 which give

the following results 3.3 and 3.4respectively.

3.3. Result: The chromatic polynomial of a total graph of a graph of order two with size one is ( -1)[( -

2)p+q-2] .

V

Figure 9: A Graph G of Order 2 Size 1 with its Total Graph

Consider a total graph T(G) as in figure 9, and we apply deletion contraction algorithm to T(G)

,the total graph can be reduced in to null graphs ,then by the theroem2.1we get the chromatic polynomial

of a total graph T(G).That is = N3 - N2 - N2 + N1 - N¬2 + N1= 3 - 3 2 +2 . The process of getting

chromatic polynomial is explained in figure10.Also this chromatic polynomial is equal to the

polynomial obtained from the analysis of structure of the graph P(T(G), ).In general a graph with p

vertices and q edges the chromatic polynomial of its chromatic polynomial is ( -1)[( -2)p+q-2] if p + q <

4,where p 3, 1 q 3.

G T(G)

e

e

v2

v1 v2

v1

R.V.N.Srinivasarao,J.Venkateswararao & D.V.S.R.Anilkumar 100

Figure 10: The Deletion -Contraction Algorithm of Total Graph T(G)

RESULT

The chromatic polynomial of a total graph of a graph of order 3and size 2 is ( -1)[( -2)3+ ( -2)] ,in general ( -1)[( -2)p+q-2+ ( -2)p+q-4].

PROOF

We have prove this result by consider the Total graph T(G)of a (3,2)-connected graph G as in

figure11 and the figure 12 explicate the procedure to establish the chromatic polynomial of graph given

in the figure 11.Here we use the theorem 2.6,which is an application of deletion –contraction algorithm

to established the chromatic polynomial of Total graph .

Figure11: A Graph G of Order 3 with its Total Graph T(G)

V1 V2 V3 e1 e2 V1

V2

V2

e1 e2

G T(G)

101 Construction of Chromatic Polynomials on Total Graphs

Figure 12: Deletion-Contraction Algorithm of T(G)

= K5 + K4 + K4 + K3 + K4+ K3

= K5+3 K4+2 K3

= λ 5-7λ 4+19λ 3-23λ 2+10λ .

= λ ( λ -1)[( λ -2)3+ ( λ -2)]

= λ ( λ -1)[( λ -2)p+q-2+ ( λ -2)p+q-4](in general).

P(T(G), ), the chromatic polynomial of a total graph of the given graph, where p is the number

of vertices of G and q is the number of edges of G.

RESULT

Determine the Chromatic Polynomial for Total Graph of a complete graph on three vertices.

Proof: The result 3.4 establishes the chromatic polynomial of total graph of (3, 2)-connected

graph. Now we extended for (3,3)-connected graph. Let G be a complete graph with 3 vertices, where,

V(G) = {v1, v2, v3}and E(G) = {e1,e2.e3} as vertices and edges respectively.

Let G be a complete graph on three vertices and T(G) is its total graph which can be shown in

figure 13. Since by its definition, the total graph contains more vertices and edges. Hence it is difficult to

find the chromatic polynomial by analysis of the structure of the graphs. Hence we use deletion-

contraction algorithm to find its chromatic polynomial. We use an equivalent form of deletion-

contraction algorithm as proved in theorem 2.6.That is if G be a graph containing nonadjacent vertices

u and v and let H be the graph obtained from G by contracting u and v. Then P(G, ) = P(G + uv, ) +

P(H, ) .

R.V.N.Srinivasarao,J.Venkateswararao & D.V.S.R.Anilkumar 102

Figure .13:A graph G with its Total Graph T(G)

In this process we add new edges between non adjacent vertices until the given graph translate

to a complete graph .Since we know that the chromatic polynomial of a complete graph on n vertices is

with given colours is ( )( -1)( -2)……( -n+1),we get the chromatic polynomial of Total graph T(G).This

can be completely explained in the figure14.Here the dotted lines represents newly added edge and the

incident vertices are contacted .Since by the theorem 2.6 and from the figure 14 we get P(G, ) = P(G +

uv, ) + P(H, ).

P(G, ) = K6+ K5+ K5+ K4+ K5+ K4+ K4+ K3

= K6+3 K5+3 K4+ K3 = 6-12 5+58 4-137 3+154 2+-64 = P(T(G), ), the chromatic

polynomial of a total graph T(G) of the given graph G.Hence we acquire the chromatic polynomials of

total graphs of (p, q) - connected graphs with p 3, 1 q 3.

CHARACTERIZATIONS F CHROMATIC POLYNOMIALS

LEMMA The chromatic polynomial of a total graph is always a polynomial in

PROOF

Let G be a finite graph and T(G)its total graph. To get the chromatic polynomial of any

graph,we use the contraction deletion algorithm as in theorem 2.2.The process terminates when all of the

remaining graphs are null graphs .Since the chromatic polynomial of a null graph of order n is and we

see that the results from above section three , the chromatic polynomial of T(G )is equal to the sum of a

large number of polynomials in and must itself be a polynomial in . Hence chromatic polynomial of a

total graph is a polynomial in .

REMARKS

As we compare the chromatic polynomial of the graphs we notice that they have some

interesting properties P(N3, )= 3, P(k3, )= 3-3 3+2 ,

P(P3, )= 3-2 3+ .

103 Construction of Chromatic Polynomials on Total Graphs

There is no constant term in each of the polynomials. Hence zero is always a root for the

chromatic polynomial.

As we observed, the sum of the coefficients of each polynomial is zero (except for null graph)

hence 1 is always a root for the chromatic polynomial. Hence, any graph with more than 1 vertex and at

least one edge cannot be properly coloured with only one colour.

The absolute value of the coefficient on the term n-1 is the number of edges of the graph.

Figure 14: Deletion-Contraction Algorithm of T(G)

THEOREM

Let G be a (p, q) - connected graph and T(G) its total graph of order n and size m where n = p +

q. Then P (T (G), ) is a monic polynomial of degree n such that the coefficient of n-1 is - m, and whose

coefficients alternate in sign.

R.V.N.Srinivasarao,J.Venkateswararao & D.V.S.R.Anilkumar 104

PROOF

We prove this property by induction on m. If m = 0, then G = and P (T(G), ) = n, as we have

seen. Then P ( , ) = n has the desired properties.

Assume that the result holds for all graphs whose size is less than m, where m ≥ 1. Let T(G) be

a graph of order m and let e = uv an edge of T(G). By theorem 2.1, P(G, ) = P(G - e, ) - P(G/e, ), Where

G/e is the graph obtained from T(G) by contract u and v. Since T(G) - e has order n and size m - 1, it

follows by the induction hypothesis that

P( T(G) - e, ) = ao n +al n-l + a2 n-2 + ... + an-l + an,

Where a0 = 1, a1 = - (m-1), ai ≥0 if i is even with 0≤ i ≤ n, ai≤0 if i is odd with 1≤ i ≤ n.

Furthermore, since H has order n-1 and size , where ≤ m-1, it follows that

P(T(G)/e, ) = bo n-1 + bl n-2 + b2 n-3 + ... + bn-2 + bn-1,

Where b0=1, b1 = - , bi ≥ o if i is even with 0≤ i ≤ n-1, and bi ≤0 if i is odd with 0≤ i ≤ n

-1. By deletion contraction theorem

P(T(G),) = P(T(G) - e, ) - P(H,)

= (ao λ n + al λ n-1` + a2 λ n-2 + ... + an-1 λ + an)-

(bo λ n-1 + bl λ n-2 + b2 λ n-3 + ... + bn-2 λ + bn-1)

= ao λ n + (al-b0) λ n-1 + (a2-b1) λ n-2 +…..

+ (an-1 – bn-2)λλλλ + (an –bn-1) Since a0 = 1, a1 –b0 = -(m-1)-1 = - m, ai – bi-1≥0 if i is even with 2≤ i ≤ n, and ai –bi-1 ≤ 0 if i

is odd with 0≤ i ≤ n, P(T(G),) has the desired properties and the theorem follows by mathematical

induction. Since the leading coefficient of the chromatic polynomial is 1, hence every chromatic

polynomial is always monic polynomial.

APPLICATIONS OF CHROMATIC POLYNOMIALS

The chromatic polynomial is to help to solve scheduling conflicts. For example, given a set of

jobs, some of which cannot be done at the same time, we can find the least amount of the time that is

needed to complete all the tasks. We will let each job be represented by a vertex, and two vertices are

adjacent to each other if the two jobs cannot to be done concurrently. Finding the chromatic polynomial

of this graph would then give us the solution to this problem.

105 Construction of Chromatic Polynomials on Total Graphs

CONCLUSIONS

Consequently, in this manuscript we found the chromatic polynomial of various graphs using

deletion and contraction algorithm. We also extend the concept to the total graphs and find the chromatic

polynomials of total graphs of (p, q)-connected graphs with p 3, 1 q 3.We also discuss the various

algebraic properties of total chromatic polynomials.

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