ON CHROMATIC D-POLYNOMIAL OF GRAPHS

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Smitha Rose, Ida David, Sudev Naduvath

To cite this version:

Smitha Rose, Ida David, Sudev Naduvath. ON CHROMATIC D-POLYNOMIAL OF GRAPHS.

CONTEMPORARY STUDIES IN DISCRETE MATHEMATICS, CENTRE FOR STUDIES IN DIS-

CRETE MATHEMATICS, 2018, 2 (1), pp.31-43. �hal-01906964�

C ONTEMPORARY S ^{TUDIES IN} D ^ISCRETE M ^ATHEMATICS

http://www.csdm.org.in

O N CHROMATIC D-P OLYNOMIAL OF GRAPHS

SMITHAROSE1,∗, IDADAVID2ANDSUDEVNADUVATH3 1Department of Mathematics, Christ University, Bengaluru, India.

2,3Department of Mathematics, Vidya Academy of Science & Technology, Thrissur, India.

Received: 15 July 2017, Revised: 12 December 2017, Accepted: 05 February 2018 Published Online: 23 February 2018.

Abstract: In graph theory, graph coloring is a special case of graph labeling; it is an assignment of labels, colors or weights to elements of a graph subject to certain constraints. Coloring the vertices of a graph in such a way that adjacent vertices are having different colors is called proper vertex coloring. A proper vertex coloring using minimum parameters of colors is studied extensively in recent literature. In this paper, we define new polynomials called chromatic D-polynomial and modified chromatic D-polynomial in terms of minimal parameter coloring and structural characteristics of graphs such as distances and degrees of vertices.

Keywords: Graph colouring, chromaticD-Polynomial.

Mathematics Subject Classification 2010: 05C12, 05C15, 05C31, 05C75.

1 Introduction

The field of graph colouring has been a fascinating research area for mathematicians since the second half of 19^th century. Many well known problems and conjectures are being studied and solved related to this subfield of Graph Theory. Inter-related research of graph colouring with its many other subfields are massive and highly application oriented. Mathematical Chemistry, biological networks, optimization and distribution networks are some areas of its applications.

Graph colouring is an assignment of colours, labels or weights to the elements of graphs subject to certain conditions. For all terms and definitions, not defined specifically in this paper, we refer to [1,2,5,11]. Moreover, for notions and norms in graph colouring, see [3,6,7]. Unless mentioned otherwise, all graphs considered here are undirected, simple, finite and connected. Also, graph colouring denotes vertex colouring in this paper.

The notion ofdistance/geodesic distance in graph theory pertains to the number of edges in the shortest path between two vertices of a graph. A graph that always has a path between any pair of its vertices is considered to be connected. If the two vertices concerned belong to different connected components of a graph, the distance is conventionally defined to be infinite. For an undirected graph, it can be noted that the vertex set and the distance function form a metric space if and only if the graph is connected.

A proper vertex colouring of a graph G is an assignment ϕ:V(G)→C of the vertices of G, where C = {c₁,c₂,c₃, . . . ,c_`}is a set of colours such that adjacent vertices ofGare assigned with different colours. The cardinality of the minimum set of colours which allows a proper colouring of G is called the chromatic number of G and is denoted χ(G). The set of all vertices ofGwhich have the colourc_i is called thecolour class of that colourc_i inG.

The cardinality of the colour class of a colourc_iis said to be thestrengthof that colour inGand is denoted byθ(c_i).

We define a functionζ :V(G)→ {1,2,3, . . . , `}such thatζ(v<sub>i</sub>) =sif and only ifϕ(v<sub>i</sub>) =c<sub>s</sub>,c<sub>s</sub>∈C, where`=χ(G) A vertex colouring consisting of the colours with minimum subscripts may be called a minimum parameter colouring (see [8]). If we colour the vertices ofGin such a way that c₁is assigned to a maximum possible number of vertices, then c₂ is assigned to a maximum possible number of remaining uncoloured vertices and proceed in this manner until all vertices are coloured, then such a colouring is called a χ⁻-colouring ofG (see Kok2). In a similar manner, if c<sub>`</sub> is assigned to a maximum possible number of vertices, then c`−1 is assigned to a maximum possible

∗ c 2018 Authors

number of remaining uncoloured vertices and proceed in this manner until all vertices are coloured, then such a colouring is called aχ⁺-colouring ofG.

Motivated by the studies mentioned above, in this paper, we define a new polynomial called chromatic D- polynomials and discuss this polynomial in terms of certain structural properties of graphs.

2 Chromatic D- Polynomial of Graphs

The term chromaticD-Polynomial of a graph, with respect to a proper colouring ofG, is be defined as follows:

Definition 1.LetGbe a connected graph with chromatic number χ(G), then with respect to a proper colouringC = {c₁,c2, . . . ,c_χ(G)}ofG, and for real variablesxandy, thechromatic D-PolynomialofG, denoted byDχ(G,x,y)is defined as

D_χ(G,x,y) =

∑

vi,vj∈V(G)

d(v_i,v_j)x^ζ(vi)y^ζ(vj),i<j.

In view of Definition1, the two types of chromaticD-polynomials we discuss in this paper, are defined as given below.

Definition 2.LetGbe a connected graph andχ⁻andχ⁺be the minimal and maximal parameter colouring ofG. Then, for real variablesxandy,

(i) theχ⁻-chromatic D-PolynomialofG, denoted byD_χ⁻(G,x,y), is defined as D_χ⁻(G,x,y) =

∑

vi,vj∈V(G)

d(v_i,v_j)x^ζχ−(vi)y^ζχ−(vj);

(ii) theχ⁺-chromatic D-PolynomialofG, denoted byD_χ+(G,x,y), is defined as D_χ+(G,x,y) =

∑

v_i,v_j∈V(G)

d(v_i,v_j)x^ζχ+(vi)y^ζχ+(vj),x,y∈R;

2.1 Chromatic D-Polynomials of Paths and Cycles

The following results determine the two types of chromaticD-Polynomials of certain graph classes.

Theorem 1.LetP_n be a path onnvertices. Then we have,

D_χ−(P_n,x,y) =











n 2

∑

i=1

(2i−1)h_n−2(i−1)

2 xy²+ⁿ⁻²ⁱ₂ x²y i

+i

(n−2i)(xy+x²y²)

; ifn is even

n−1 2

∑

i=1

(2i−1)h_n−(2i−1)

2 (xy²+x²y)i +i

(n−(2i−1))xy+ (n−(2i+1))x²y²

; ifn is odd;

Proof. LetV ={v₁,v2, . . . ,vn} be the ordered vertex set of Pn, with χ(P_n) =2. Since Pn has diameter n−1, the distanced(v_i,v_j)can vary from1ton−1. Now consider the following two cases:

Case-1:Letnbe even. Then, with respect to aχ⁻-colouring, the verticesv₁,v₃,v₅, . . .vn−1are coloured with the colourc₁and the verticesv₂,v₄,v₆, . . .v_nget the colourc₂(see Figure given below).

c₁ c₂ c₁ c₂ c₁ c₂ c₁ c₂

Fig. 1:The graphP₈withχ⁻-colouring

The possible colour pairs and their numbers inGin terms of the distances between them are listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs r even (c₁,c₁) ^n−r₂

(c₂,c2) ^n−r₂ r odd (c₁,c₂) ^n−(r−1)₂

(c₂,c₁) ^n−(r+1)₂ Table 1

From Table-1, we have

D_χ−(P_n,x,y) = ∑

r even

rn−r

2 (xy+x²y²) + ∑

rodd

rh_n−(r−1)

2 xy²+^n−(r+1)₂ x²yi Further simplifications gives the result as

D_χ⁻(P_n,x,y) =

n 2

∑

i=1

(2i−1)

n−2(i−1)

2 xy²+n−2i 2 x²y

+i

(n−2i)(xy+x²y²) .

Case-2:Letnbe odd. Then, with respect to aχ⁻-colouring, the verticesv₁,v₃,v₅, . . .v_nget the colourc₁and the verticesv₂,v₄,v₆, . . .vn−1 get the colourc₂. (see Figure given below).

c₁ c₂ c₁ c₂ c₁ c₂ c₁ c₂ c₁

Fig. 2:The graphP₉withχ⁻-colouring

The possible colour pairs and their numbers inGin terms of the distances between them are listed in the following table.

Distanced(v_i,vj) Colour pairs Number of pairs r even (c₁,c₁) ^n−(r−1)₂

(c₂,c₂) ^n−(r+1)₂ r odd (c₁,c₂) ^n−r₂

(c₂,c₁) ^n−r₂ Table 2

From Table-2, it is noted thatD_χ⁻(P_n,x,y) = ∑

reven

rh_n−(r−1)

2 xy+^n−(r+1)

2 x²y²i + ∑

rodd

r_n−r

2 (xy²+x²y) . On simplification, we getD_χ−(P_n,x,y) =

n−1 2

∑

i=1

(2i−1)h_n−(2i−1)

2 (xy²+x²y)i +i

(n−(2i−1))xy+ (n−(2i+1))x²y² . This completes the proof.

In the following theorem, we obtain the results for chromaticD-Polynomial of paths using χ⁺ colouring as follows:

Theorem 2.LetPn be a path onnvertices. Then we have,

D_χ+(P_n,x,y) =











n 2

∑

i=1

(2i−1)h

n−2i

2 xy²+^n−2(i−1)₂ x²yi +i

(n−2i)(xy+x²y²)

; ifn is even

n−1 2

∑

i=1

(2i−1)h_n−(2i−1)

2 (xy+x²y²)i +i

(n−(2i+1))xy²+ (n−(2i−1))x²y

; ifn is odd;

Proof. Let P_n be a path with the ordered vertex setV ={v₁,v₂, . . . ,v_n}. Since P_n has diameter n−1, the distance d(v_i,vj)can vary from1 ton−1. Now consider the following two cases:

Case-1:Letnbe even. Then, with respect to aχ⁺-colouring, the verticesv₁,v₃,v₅, . . .vn−1are coloured with the colourc₂and the verticesv₂,v₄,v₆, . . .v_nget the colourc₁. The possible colour pairs and their numbers inGin terms of the distances between them are listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs r even (c₂,c2) ^n−r₂

(c₁,c₁) ^n−r₂ r odd (c₂,c1) ^n−(r−1)₂

(c₁,c₂) ^n−(r+1)₂ Table 3

From Table-3, we haveD_χ+(P_n,x,y) = ∑

r even

r_n−r

2 (xy+x²y²) + ∑

rodd

rh_n−(r−1)

2 x²y+^n−(r+1)₂ xy²i . On further simplification, we getD_χ+(P_n,x,y) =

n 2

∑

i=1

(2i−1)h_n−2(i−1)

2 x²y+ⁿ⁻²ⁱ₂ xy²i +i

(n−2i)(xy+x²y²) . Case-2:Letnbe odd. Then, with respect to aχ⁺-colouring, the verticesv₁,v₃,v₅, . . .v_nget the colourc₂and the verticesv2,v4,v₆, . . .v_n−1 get the colourc1(see Figure below).

c₂ c₁ c₂ c₁ c₂ c₁ c₂

Fig. 3:The graphP₇withχ⁺-colouring

The possible colour pairs and their numbers inGin terms of the distances between them are listed in the following table. From Table-4, we haveD_χ⁺(P_n,x,y) = ∑

r even

r

h_n−(r−1)

2 x²y+^n−(r+1)₂ xy² i

+ ∑

r odd

rn−r

2 (xy+x²y²) .

Distanced(v_i,v_j) Colour pairs Number of pairs r even (c₂,c₁) ^n−(r−1)₂

(c₁,c₂) ^n−(r+1)₂ r odd (c₂,c₂) ^n−r₂

(c₁,c₁) ^n−r₂ Table 4

That is,D_χ+(P_n,x,y) =

n−1 2

∑

i=1

(2i−1)h_n−(2i−1)

2 (xy+x²y²)i +i

(n−(2i−1))x²y+ (n−(2i+1))xy² . This completes the proof.

The following two theorems discuss the chromaticD-Polynomial of cycles with χ⁻ andχ⁺colouring.

Theorem 3.LetC_nbe a cycle on nvertices. Then we have,

Dχ−(C_n,x,y) =



















n−2 2

∑

i=1

ni

xy+x²y²

+ (2i−1)_n+2

2 xy²+ⁿ⁻²₂ x²y + ∑

i=ⁿ₂

i_i

2(xy+x²y²) +ⁱ⁺¹₂ xy²+ⁱ⁻¹₂ x²y

; if nis even (4i−1)(xy³+x²y³) +

n 2

∑

i=1

2ih_n−(2i+1)

2 (xy+x²y²) +ixy²+²ⁱ⁻¹₂ x²yi + (2i−1)h_n−(2i−1)

2 xy²+^n−(2i+3)₂ x²y+ (i−1)(xy+x²y²)i

; if nis odd;

Proof. LetV={v₁,v₂, . . . ,v_n}be the ordered vertex set ofC_nin clockwise direction. Now consider the following two cases:

Case-1: Let n be even. We have χ(C_n) =2 and diameter is ⁿ₂. So the distance d(v_i,vj) can vary from 1 to ⁿ₂. Then, with respect to a χ⁻-colouring, the verticesv₁,v₃,v₅, . . .vn−1 are coloured with the colourc₁and the vertices

v₂,v₄,v₆, . . .v_nget the colourc₂. The possible colour pairs and their numbers inGin terms of the distances between

them are listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs r is even, n=2r (c₁,c₁) ^r

(c₂,c₂) 2₂^r

r is even,n>2r (c₁,c₁) ⁿ₂ (c₂,c₂) ⁿ₂ r is odd, n=2r (c₁,c₂) ^r+1₂

(c₂,c1) ^r−1₂ r is odd,n>2r (c₁,c2) ⁿ⁺²₂ (c₂,c₁) ⁿ⁻²₂ Table 5:Whennis even

From Table-5, when n is even, we have Dχ−(C_n,x,y) =^n>2r∑

reven

r(ⁿ₂(xy+x²y²)) +^n=2r∑

reven r

2(xy+x²y²) +^n>2r∑

rodd

r(ⁿ⁺²₂ xy²+

n−2

2 x²y) +^n=2r∑

rodd

r(^r+1₂ xy²+^r−1₂ x²y). Further simplifications will give the result as:

D_χ⁻(C_n,x,y) =

n−2 2

i=1

∑

ni

xy+x²y²

+ (2i−1) n+2

2 xy²+n−2 2 x²y

+

∑

i=ⁿ₂

i i

2(xy+x²y²) +i+1

2 xy²+i−1 2 x²y

.

Case-2:Letnbe odd. We haveχ(C_n) =3 and diameter isbⁿ₂c. So the distanced(v_i,v_j)can vary from1to ⁿ⁻¹₂ . Then, with respect to the conditions as per aχ⁻-colouring, the verticesv₁,v₃,v₅, . . .vn−2are coloured with the colour c1 and the vertices v2,v4,v₆, . . .v_n−1 get the colour c2. Also, v_n is coloured with c3. The possible colour pairs and their numbers inC_nin terms of the distances between them are listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs

r is even

(c₁,c₁) ^n−r−1₂ (c₂,c2) ^n−r−1₂ (c₁,c₂) ₂^r (c₂,c₁) ^r−2₂ (c₁,c₃) 1 (c₂,c3) 1

r is odd

(c₁,c1) ^n−r₂ (c₂,c₂) ^n−r−2₂ (c₁,c₂) ^r−1₂ (c₂,c₁) ^r−1₂ (c₁,c₃) 1 (c₂,c3) 1 Table 6:Whennis odd

Invoking the definition of chromaticD-Polynomials, when n is odd, from Table-6, and we have D_χ−(C_n,x,y) =

∑

reven

r_n−r−1

2 (xy+x²y²) +^r₂xy²+^r−2₂ x²y+ (xy³+x²y³) + ∑

rodd

r_n−r

2 xy²+^n−r−2₂ x²y+^r−1₂ (xy+x²y²) + (xy³+x²y³) .

Further simplifications will give the result as follows: D_χ⁻(C_n,x,y) =

n 2

∑

i=1

2ih_n−(2i+1)

2 (xy+x²y²) +ixy²+²ⁱ⁻¹₂ x²yi + (2i−1)h_n−(2i−1)

2 xy²+^n−(2i+3)₂ x²y+ (i−1)(xy+x²y²)i

+ (4i−1)(xy³+x²y³).

v₁ c₁

v₂ c₂ v₃ c1

v₄ c₂ v₅ c₁

v₆ c2 v7

c₁

v₈ c2

v₉ c3

v₈ c₂

v₇ c₁

v₆ c2

v₅ c1

v4

c₂ v₃ c1

v₂ c₂ v₁ c₁

Fig. 4:χ−colouring ofC8andC9

Theorem 4.LetC_nbe a cycle on nvertices. Then we have,

D_χ+(C_n,x,y) =



















n−2 2

∑

i=1

ni

xy+x²y²

+ (2i−1)_n−2

2 xy²+ⁿ⁺²₂ x²y + ∑

i=ⁿ₂

i_i

2(xy+x²y²) +ⁱ⁻¹₂ xy²+ⁱ⁺¹₂ x²y

; ifnis even

n 2

∑

i=1

(4i−1)(x³y+x²y) + (2i−1)h_n−(2i−1)

2 x³y²+^n−(2i+3)₂ x²y³+ (i−1)(x³y³+x²y²)i + 2ih_n−(2i+1)

2 (x³y³+x²y²) +ix³y²+²ⁱ⁻¹₂ x²y³i

; ifnis odd.

Proof. LetV={v₁,v2, . . . ,vn}be the ordered vertex set ofCnin clockwise direction. Now consider the following two cases:

Case-1: Let n be even. Here the chromatic number and the diameter are χ(C_n) =2 and ⁿ₂ respectively. The distanced(v_i,vj)here vary from1to ⁿ₂. To imply aχ⁺-colouring, the verticesv1,v3,v₅, . . .vn−1are coloured with the colourc₂and the verticesv₂,v₄,v₆, . . .v_nget the colourc₁. (See the figure5)The following table describes the colour pairs in accordance with the distance as:

Distanced(v_i,v_j) Colour pairs Number of pairs r is even, n=2r (c₁,c₁) ^r₂

(c₂,c₂) ^r₂ r is even,n>2r (c₁,c₁) ⁿ₂ (c₂,c₂) ⁿ₂ r is odd, n=2r (c₁,c₂) ^r−1₂

(c₂,c₁) ^r+1₂ r is odd,n>2r (c₁,c₂) ⁿ⁻²₂ (c₂,c₁) ⁿ⁺²₂ Table 7:Whennis even

From Table-7, whenn is even, we have

D_χ+(C_n,x,y) =

n>2r reven∑

r(ⁿ₂(xy+x²y²)) +

n=2r reven∑

r

2(xy+x²y²) +

n>2r

∑

rodd

r(ⁿ⁻²₂ xy²+ⁿ⁺²₂ x²y) +^n=2r∑

rodd

r(^r−1₂ xy²+^r+1₂ x²y) Further simplifications will give the result as:

D_χ⁺(C_n,x,y) =

n−2 2

i=1

∑

ni

xy+x²y²

+ (2i−1) n−2

2 xy²+n+2 2 x²y

+

∑

i=ⁿ₂

i i

2(xy+x²y²) +i−1

2 xy²+i+1 2 x²y

.

Case-2: Let n be odd. As in the previous theorem, the chromatic number is χ(C_n) =3 and diameter is bⁿ₂c.

Applying the colouring as in figure5the distances and colour pairs are analysed as per the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs

r is even

(c₃,c3) ^n−r−1₂ (c₂,c₂) ^n−r−1₂ (c₃,c₂) ₂^r (c₂,c₃) ^r−2₂ (c₃,c₁) 1 (c₂,c₁) 1

r is odd

(c₃,c₃) ^n−r₂ (c₂,c₂) ^n−r−2₂ (c₃,c₂) ^r−1₂ (c₂,c₃) ^r−1₂ (c₃,c₁) 1 (c₂,c₁) 1 Table 8

Then, we have D_χ+(C_n,x,y) = ∑

r odd

r_n−r

2 x³y²+^n−r−2₂ x²y³+^r−1₂ (x³y³+x²y²) + (x³y+x²y) +

∑

r even

r_n−r−1

2 (x³y³+x²y²) +^r₂x³y²+^r−2₂ x²y³+ (x³y+x²y) , On simplification, D_χ+(C_n,x,y) =

n 2

∑

i=1

(4i−1)(x³y+x²y) +2ih_n−(2i+1)

2 (x³y³+x²y²) +ix³y²+²ⁱ⁻¹₂ x²y³i

+ (2i− 1)h_n−(2i−1)

2 x³y²+^n−(2i+3)₂ x²y³+ (i−1)(x³y³+x²y²)i .

v1 c2

v2

c₁ v₃ c₂ v4

c₁ v₅ c2

v₆ c₁ v7

c₂

v₈ c₁

v₉ c1

v₈ c₂

v₇ c₃

v₆ c2

v₅ c3

v4

c₂ v₃ c3

v₂ c₂ v₁ c₃

Fig. 5:χ+colouring ofC₈andC₉

2.2 Chromatic D Polynomials of Complete Graphs and Complete Bipartite Graphs

In the following theorem we obtain the results for chromatic D-Polynomial of complete graphs using χ⁻ and χ⁺ colouring as follows:

Theorem 5.LetK_nbe a complete graph onn vertices. Then we have, (i)D_χ−(K_n,x,y) =

n−1

∑

i=1

xⁱyⁱ⁺¹+xⁿy (ii)D_χ+(K_n,x,y) =

n−1

∑

i=1

xⁱ⁺¹yⁱ+xyⁿ

Proof. Consider the complete graphK_nonnvertices. Since all pairs of vertices of this graph are adjacent, there are ncolour classes of strength1 each and the maximum possible distance is1.

Part (i): According to the minimum parameter colouring the ordered colour set C ={c1,c2,c3, . . . ,c_n} is being used for colouring the vertices of a complete graph such that the vertexvn assumes the colour cn, for 1≤i≤n.

Any edge v_iv_j will have the colour codes c_ic_i+1, except for the last edge v_nv₁. From the definition of chromatic D-Polynomial, the result follows as:

D_χ−(K_n,x,y) =

n−1 i=1

∑

xⁱyⁱ⁺¹+xⁿy .

Part (ii):In order to get the maximum values of chromaticD-Polynomial using the minimum parameter colouring the ordered colour setC={c_n,cn−1,cn−2, . . . ,c₁}is used for colouring the vertices of a complete graph such that the vertexv_n assumes the colourcn−i, for0≤i≤n−1. Any edgev_iv_j will have the colour codes c_i+1c_i, except for the last edgev₁v_n. From the definition of chromaticD-Polynomial, the result follows as:

D_χ+(K_n,x,y) =

n−1 i=1

∑

xⁱ⁺¹yⁱ+xyⁿ .

Next, in this section, the two types of chromaticD-Polynomials of complete bipartite graphs are being discussed.

Theorem 6.LetKm,nbe a complete bipartite graph onm+nvertices. Then we have,

Dχ−(K_m,n,x,y) =

(m²xy²+m(m−1)(xy+x²y²); if m=n mnxy²+ⁿ⁽ⁿ⁻¹⁾

2 xy+^m(m−1)

2 x²y²; if m<n;

Proof. Consider the complete bipartite graph K_m,n withm≤n. We have two independent partite setsA,B and for the easy flow of the proof we fix their cardinalities as|m|,|n|respectively and carefully order the vertices such that A and B have ordered vertex sets {v_n+1,v_n+2, . . . ,v_n+m} and {v₁,v₂, . . . ,v_n} respectively. Thus V ={v₁,v₂, . . . ,v_n,vn+1,vn+2, . . . ,vn+m} forms the ordered vertex set of the bipartite graph Km,n, with χ(K_m,n) =2.

SinceKm,n has diameter2, the distanced(v_i,vj)withi<j can vary between1or2. We now consider the following two cases:

Case-1:Let we have,m=n. Then, with respect to a χ⁻-colouring, the verticesv₁,v₂,v₃, . . .v_m are coloured with the colourc1 and the verticesvm+1,v_m+2, . . .v_2m get the colourc2. The possible colour pairs and their numbers in Km,nin terms of the distances between them are listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs

1 (c₁,c₂) m²

2 (c₁,c₁) ^m(m−1)₂

(c₂,c₂) ^m(m−1)₂ Table 9:Table ofKm,nwhenm=n

From Table-9, we have

Dχ−(K_m,n,x,y) =m²xy²+m(m−1)(xy+x²y²)

Case-2:Letm<nbe assumed. Here the vertices of setAandBare coloured withc₂andc₁respectively. Thus the ordered vertex sets{v₁,v2, . . . ,v_n} and{v_n+1,vn+2, . . . ,vn+m} gets colours c2 andc1 respectively. The possible colour pairs and their numbers inK_m,nin terms of the distances between them are listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs

1 (c₁,c₂) mn

2 (c₁,c1) ⁿ⁽ⁿ⁻¹⁾₂

(c₂,c₂) ^m(m−1)₂ Table 10:Table ofKm,nwhenm<n

From Table-10, we have

D_χ−(K_m,n,x,y) =mnxy²+n(n−1)

2 xy+m(m−1) 2 x²y²

This completes the proof.

v₁ c₁

v₂ c₁

v₃ c₁

v₄ c₁ v₅

c₂

v₆ c₂

v7

c₂

Fig. 6:The graphK3,4

In the following theorem we obtain the results for chromaticD-Polynomial of complete bipartite graphs usingχ+

colouring as follows:

Theorem 7.LetK_m,nbe a complete bipartite graph onm+nvertices. Then we have,

D_χ+(K_m,n,x,y) =

(m²x²y+m(m−1)(xy+x²y²); if m=n mnx²y+ⁿ⁽ⁿ⁻¹⁾₂ x²y²+^m(m−1)₂ xy; if m<n;

Proof. In χ+colouring, we colour the sets Awith c₁ andBwith c₂ and the procedure of the proof is more or less similar to the proof of the Theorem 6

2.3 Chromatic D-Polynomials of Wheel and Double Wheel Graphs

In the following theorems, the results for chromatic D-Polynomial of the wheel and the double wheel graphs using χ⁻ andχ⁺colouring are worked out:

Theorem 8.For a wheelW_n=C_n+K₁, we have

Dχ−(W_n,x,y) =







(n+2)(n−2)

4 xy²+⁽ⁿ⁻²⁾₄ ²x²y+ⁿ⁽ⁿ⁻²⁾₄ (xy+x²y²) + ⁿ₂(xy³+x²y³); ifnis even;

(n−1)²

4 xy²+⁽ⁿ⁻¹⁾₂ (xy³+xy⁴+x²y⁴) +⁽ⁿ⁻³⁾₄ ²x²y+^{(n−1)(n−3)}₄ (xy+x²y²)

+⁽ⁿ⁻³⁾₂ (x²y³+x³y⁴); ifnis odd.

c1 v1

c2 v2

c₁ v3

c2

v4

c1

v₅ c₂ v₆

c₁ v₇

c₂ v₈

c₃ v₉ c1 v1

c₂ v₂ c₁

v3

c₂ v₄ c₁ v₅

c₂ v₆

c₁ v₇

c₂ v₈ c₃ v₉

c4

v₁₀

Fig. 7:χ⁻- colouring of wheel graphsW8andW9

Proof. Consider the wheel graphW_n=C_n+K₁ on n+1 vertices. Letv₁,v₂,· · ·v_n be the vertices ofC_non the rim of the wheel and vn+1 be the central vertex. It is obvious that, W_n has chromatic number 3 when n is even and chromatic number4 whennis odd. SinceW_nhas diameter2, the distance d(v_i,v_j)can be1 or2. Now consider the following cases:

Part (i): Whennis even.Then according to the rules ofχ⁻-colouring, the verticesv₁,v₃,v₅, . . .vn−1are coloured with the colourc₁and the vertices v₂,v₄,v₆, . . .v_nget the colourc₂. The central vertexv_n+1will be coloured withc₃. The possible colour pairs with appropriate distances and their numbers inW_nare listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs 1

(c₁,c₂) ⁿ⁺²

2

(c₂,c₁) ⁿ⁻²₂ (c₁,c₃) ⁿ₂ (c₂,c₃) ⁿ₂

2

(c₁,c1) ⁿ⁽ⁿ⁻²⁾₈ (c₂,c2) ⁿ⁽ⁿ⁻²⁾₈ (c₁,c2) ^(n−4)(n+2)₈ (c₂,c1) ^{(n−4)(n−2)}₈ Table 11:Whennis even

From Table-11, when nis even, Dχ−(W_n,x,y) =n+2

2 xy²+n−2 2 x²y+n

2xy³+n

2x²y³+n(n−2)

4 (xy+x²y²) +(n−4)(n+2)

4 xy²+(n−4)(n−2) 4 x²y Further simplifications will give the result as follows:

D_χ−(W_n,x,y) =(n+2)(n−2)

4 xy²+(n−2)²

4 x²y+n(n−2)

4 (xy+x²y²) +n

2(xy³+x²y³)

Part (ii): When nis odd.As per the conditions of theχ⁻- colouring, the verticesv₁,v₃,v₅, . . .v_n−2are coloured with the colour c1and the verticesv2,v4,v6, . . .v_n−1get the colour c2. The vertexvnwill be coloured c3 and the central vertexv_n+1will be coloured withc₄. The following table analyses the possible distances in terms of the colour pairs required.

Distanced(v_i,v_j) Colour pairs Number of pairs

1

(c₁,c₂) ⁿ⁻¹₂ (c₂,c₁) ⁿ⁻³₂ (c₁,c4) ⁿ⁻¹₂ (c₂,c₄) ⁿ⁻¹₂ (c₁,c₃) 1 (c₃,c₄) 1

2

(c₁,c₁) ^{(n−1)(n−3)}₈ (c₂,c₂) ^{(n−1)(n−3)}₈ (c₁,c₂) ^{(n−3)(n−5)}₈ (c₂,c₁) ^{(n−3)(n−5)}₈ (c₁,c₃) ⁿ⁻³₂ (c₂,c₃) ⁿ⁻³₂ Table 12

From the table 12, we have D_χ−(W_n,x,y) = ⁿ⁻¹₂ (xy²+xy⁴+x²y⁴) +ⁿ⁻³₂ x²y+xy³+x³y⁴+^{(n−1)(n−3)}₄ (xy+xy²+ x²y²) +^{(n−3)(n−5)}₄ x²y+ⁿ⁻³₂ (xy³+x²y³).

O further simplification, D_χ−(W_n,x,y) =⁽ⁿ⁻¹⁾₄ ²xy²+⁽ⁿ⁻¹⁾₂ (xy³+xy⁴+x²y⁴) +⁽ⁿ⁻³⁾₄ ²x²y+^{(n−1)(n−3)}₄ (xy+x²y²) +

(n−3)

2 (x²y³+x³y⁴). This completes the proof.

Theorem 9.For a wheelW_n=C_n+K₁, we have

D_χ+(W_n,x,y) =







(n+2)(n−2)

4 x³y²+⁽ⁿ⁻²⁾²

4 x²y³+ⁿ⁽ⁿ⁻²⁾

4 (x³y³+x²y²) +ⁿ

2(x³y+x²y), ifn is even;

(n−1) 2

h_(n−1)

2 x²y³+⁽ⁿ⁻³⁾₂ x³y³(xy+1) +x²y(y+x+x²)i

+ⁿ⁻³_{2 n−3}

2 x³y²+x²y(y+1)

, ifn is odd.

Proof. In the χ− colouring of a wheel Wn of even order, if we interchange the colours c1 and c2, we get its χ+

colouring. But in the odd case, we interchange the colours c₁ withc₃. Refer to8for colouring pattern.

Part (i): When n is even.The possible colour pairs with appropriate distances and their numbers inW_n are listed in the following table.

Distanced(v_i,v_j) Colour pairs Number of pairs 1

(c₃,c₂) ⁿ⁺²₂ (c₂,c₃) ⁿ⁻²₂ (c₃,c1) ⁿ₂ (c₂,c1) ⁿ₂

2

(c₃,c₃) ⁿ⁽ⁿ⁻²⁾₈ (c₂,c₂) ⁿ⁽ⁿ⁻²⁾₈ (c₃,c₂) ^(n−4)(n+2)₈ (c₂,c₃) ^{(n−4)(n−2)}₈ Table 13:Whennis even

From Table-13, when nis even,

D_χ+(W_n,x,y) =n+2‘

2 x³y²+n−2

2 x²y³+n

2(x³y+x²y) n(n−2)

4 (x³y³+x²y²) +(n−4)(n+2)

4 x³y²+(n−4)(n−2) 4 x²y³ Further simplifications will give the result as follows:

D_χ+(W_n,x,y) =(n+2)(n−2)

4 x³y²+(n−2)²

4 x²y³+n(n−2)

4 (x³y³+x²y²)+

n

2(x³y+x²y)

Part (ii): Whennis odd.As per the conditions of theχ⁺- colouring, the verticesv₁,v₃,v₅, . . .vn−2are coloured with the colourc₄and the vertices v₂,v₄,v₆, . . .vn−1get the colour c₃. The vertexv_n will be colouredc₂ and the central vertexvn+1will be coloured withc₁. The following table analyses the possible distances in terms of the colour pairs required.

Distanced(v_i,v_j) Colour pairs Number of pairs

1

(c₄,c₃) ⁿ⁻¹₂ (c₃,c₄) ⁿ⁻³₂ (c₄,c₁) ⁿ⁻¹₂ (c₃,c1) ⁿ⁻¹₂ (c₄,c2) 1 (c₂,c₁) 1

2

(c₄,c4) ^{(n−1)(n−3)}₈ (c₃,c₃) ^{(n−1)(n−3)}₈ (c₄,c₃) ^{(n−3)(n−5)}₈ (c₃,c₄) ^{(n−3)(n−5)}₈ (c₄,c₂) ⁿ⁻³₂ (c₃,c₂) ⁿ⁻³₂ Table 14

From the table14, and the definitions of chromaticD-Polynomials it is clear that D_χ+(W_n,x,y) =n−1

2 (x⁴y³+x⁴y+x³y) +n−3

2 x³y⁴+x⁴y²+x²y+ (n−1)(n−3)

4 (x⁴y⁴+x⁴y³+x³y³) +(n−3)(n−5)

4 x³y⁴+n−3

2 (x⁴y²+x³y²) Further simplifications will give the result as follows:

Dχ−(W_n,x,y) =(n−1)²

4 x⁴y³+(n−1)

2 (x⁴y²+x⁴y+x³y) +(n−3)² 4 x³y⁴+ (n−1)(n−3)

4 (x⁴y⁴+x³y³) +(n−3)

2 (x³y²+x²y) This completes the proof.

c₃ v₁

c₂ v₂ c₃

v₃ c₂

v₄ c₃ v₅

c₂ v₆

c₃ v₇

c₂ v₈

c₁ v₉ c₄ v₁

c₃ v₂ c4

v₃ c₃ v₄ c₄ v₅

c3

v6

c₄ v₇

c₃ v₈ c₂ v₉

c₁ v₁₀

Fig. 8:χ⁺- colouring of the wheel graphsW8andW9

3 Conclusion

The study seems to be promising for further studies as the polynomial can be computed for many graph classes and classes of derived graphs. The chromaticD-Polynomial can be determined for graph operations, graph products and graph powers. The study onD-Polynomials with respect to different types of graph colourings also seem to be much promising. The concept can be extended to edge colourings and map colourings also.

These polynomials have so many applications in various fields like Mathematical Chemistry, Distribution Theory, Optimization Techniques etc. Similar studies are possible in various other fields. All these facts highlight the wide scope for further research in this area.

Acknowledgement

The first author would like to acknowledge the bits of all academic helps rendered by Centre for Studies in Discrete Mathematics, Vidya Academy of Science and Technology, Thrissur, Kerala, India.

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