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

CONSTRUCTION OF CHROMATIC POLYNOMIALS ON TOTAL GRAPHS R.V.N.SRINIVASARAO1 , J.VENKATESWARARAO2 & D.V.S.R.ANILKUMAR3 1 2 3

Department of Mathematics, Guntur Engineering College, Guntur Dt, A.P, India

Department of Mathematics, Mekelle University Main Campus, Mekelle, Ethiopia

Department 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


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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 窶田ontraction 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 窶田ontraction 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.


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

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 1.6 Example:

v1v2

e 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

χ (G ) .

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 1.11. Definition:[11]

λ

colours.

For a graph G and a positive integer λ , the number of different proper

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

χ (G ) > λ then P (G, λ )=0.

1.12.Example.If we want to colour the null graph N3 with done

λ

colours .we notice that this can be

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


95

Construction of Chromatic Polynomials on Total Graphs

位 位

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 窶田ontraction algorithm


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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 4-6 λ 3 -11 λ 2-6 λ . 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

P (C4, λ )

98

λ ( λ - 1)2+ λ ( λ -1)( λ -2)2 = λ ( λ -1)( λ 2-3 λ +3) = λ 4-4 λ 3+6 λ 2-3 λ = ( 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] .

e V

v1

e

v2

G

v1

T(G)

v2

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 - NÂŹ2 + 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.


100

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

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 窶田ontraction algorithm to established the chromatic polynomial of Total graph .

V2

V1

e1

V2

e2

V3

V2 V1

e2

e1 G

T(G)

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


101

Construction of Chromatic Polynomials on Total Graphs

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

K5 + K4

=

K5+3 K4+2 K3

=

λ 5-7 λ 4+19 λ 3-23 λ 2+10 λ .

=

λ ( λ -1)[( λ -2)3+ ( λ -2)]

=

+

K4

+

K3

+ K4+ K3

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 deletioncontraction algorithm to find its chromatic polynomial. We use an equivalent form of deletioncontraction 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)â&#x20AC;Śâ&#x20AC;Ś( -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)λ λ + (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.

REFERENCES 1.

F.M.Dong. Chromatic polynomials and chromaticity of graphs .World Scientific Publishing

Company, 2005. 2. G.D.Birkhoff, A determinant formula for the number of ways of coloring a map. Ann. Math. 14 (1912) 42-46. 3.

G.H.J.Meredith, Coefficients of Chromatic Polynomials, Journal of Combinatorial theory B13

(1972),14-17. 4. H.Whitney, The colorings of graphs. Ann. Math. 33 (1932) 688-718. 5. J.Venkateswara Rao, R.V.N.Sinivasa Rao. A unique approach on upper bounds for the chromatic number of total graphs, Asian journal of Applied Sciences 5(4) 240-246, 2012. 6. Norman Biggs. Algebraic graph theory (Cambridge mathematical Library).Cambridge University Press, February1994. 7. P.Erdos and R.J.Wilson, on the chromatic index of almost all graphs. J.Combin. Theory Ser.B 23 (1977) 255-257. 8. R.A.Brualdi, Introduction Combinatorics,3rd ed. Prentice-Hall, Upper saddle River, New Jersey1999. 9. R.C.Reed, An introduction to chromatic polynomials. J.Combin. Theory 4 (1968) 52-71. 10. Robin J.Willson, Introduction to graph theory, 1996 by Addison Wesley Publishing Company. 11. West.D.B, Introduction to Graph Theory, 2nd Edition, Prentice-Hall India edition, 2003.

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