Star Coloring of Cartesian Product of Paths and Cycles with Complete Bipartite Graphs

Star Coloring of Cartesian Product of Paths and Cycles with Complete Bipartite Graphs

P. Hemalatha and S. N. Subhathra

Vellalar College for Women, Erode 638 012, Tamilnadu, INDIA.

email:[email protected], [email protected]

(Received on: June 10, 2019) ABSTRACT

Let 𝐾𝑚,𝑛, 𝑃𝑟, 𝐶𝑟 denote a complete bipartite graph, a path on r vertices and a cycle on r vertices respectively. Let 𝐺₁, 𝐺₂ denotes the cartesian product of two graphs G1 and G2. In this paper, the star coloring of some special classes of graphs such as 𝑃𝑟, 𝐾𝑚,𝑛 and 𝐶𝑟, 𝐾𝑚,𝑛 have been considered and the star chromatic number 𝜒𝑠 is obtained for such graphs. It is proved that for given positive integers 𝑚 ≥ 3 and 𝑛, 𝑟 ≥ 2

(𝑖)𝜒𝑠(𝑃𝑟, 𝐾𝑚,𝑛) = {min{𝑚, 𝑛} + 2, 𝑖𝑓 2 ≤ 𝑟 ≤ 3 min{𝑚, 𝑛} + 3, 𝑖𝑓 𝑟 ≥ 4 (𝑖𝑖)𝜒𝑠(𝐶𝑟, 𝐾𝑚,𝑛) = {𝑚 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 2

𝑛 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 3.

Keywords: Cartesian Product of Graphs, Vertex Coloring, Star Coloring, Star Chromatic Number.

1. INTRODUCTION

All graphs considered here are finite simple undirected connected graphs.

Complete Bipartite Graph [1]

If 𝐺[𝑋, 𝑌] is simple and every vertex in X is joined to every vertex in Y, then G is called a complete bipartite graph denoted by 𝐾

, 𝑖𝑓 |𝑋| = 𝑚 and |𝑌| = 𝑛.

Path[7]

An open walk in which no vertex appears more than once is called a path. A path on r

vertices is denoted by 𝑃

.

Cycle[7]

A closed walk in which no vertex appears more than once except the initial and final vertices is called a cycle. A cycle on r vertices is denoted by 𝐶

.

Star Coloring[6]

A star coloring of a graph G is a proper coloring of G with the condition that no path on 4 vertices is 2-colored. A k-star coloring of a graph G is a star coloring of G using atmost k colors.

Cartesian Product[5]

The cartesian product of two graph G and H, denoted by 𝐺 ∈ 𝐻, is a graph with vertex set 𝑉(𝐺 ∈ 𝐻) = 𝑉(𝐺) × 𝑉(𝐻) = {(𝑔, ℎ)│𝑔 ∈ 𝐺, ℎ ∈ 𝐻}. The edge set of 𝐺 ∈ 𝐻 consists of all pairs [(𝑔

, ℎ

), (𝑔

, ℎ

)] of vertices with [𝑔

, 𝑔

] ∈ 𝐸(𝐺) 𝑎𝑛𝑑 ℎ

= ℎ

, or 𝑔

= 𝑔

and [ℎ

, ℎ

] ∈ 𝐸(𝐻).

Graph coloring is one of the best known and popular research area in graph theory.

The acyclic coloring of graphs was first introduced by Gr𝑢̈nbaum

in 1973. The acyclic coloring of planar graphs was discussed by O.V.Borodin

. The acyclic chromatic index of cartesian product of graphs was discussed by Yu-ping Tsao et al.

. Later in

, he discussed the proper coloring in which P

is not bicolored and this coloring is known as star coloring. Star coloring of graphs was discussed by G.Fertin et al.

. N.Ramya

has compared the chromatic number, acyclic coloring chromatic number and star coloring chromatic number of wheel graphs. L.Jethruth Emebla Mary et al.

has discussed the star coloring of graphs formed by the cartesian product of some simple graphs. Recently, the star coloring of cartesian product of paths and cycles have been investigated by Tianyong Han et al.

. Also, he has solved some of the SAT (satisfiability test) instances transformed from k-star coloring problems by using the software MiniSat

. All these studies motivate us to study the star coloring of graphs. In this paper, the star chromatic number of the graphs 𝑃

, 𝐾

, 𝐶

, 𝐾

have been obtained for given positive integers m, n, r ≥ 2 .

2. STAR COLORING OF CARTESIAN PRODUCT OF GRAPHS 2.1 Star Coloring of 𝑷

, 𝑲

In this section, the star coloring of the cartesian product of the path 𝑃

with the complete bipartite graph 𝐾

for given positive integers 𝑚, 𝑛, 𝑟 ≥ 2 is discussed.

Theorem 2.1. For given positive integers m ≥ 3 and n, r ≥ 2,

𝜒

(𝑃

, 𝐾

) = { min{𝑚, 𝑛} + 2, 𝑖𝑓 2 ≤ 𝑟 ≤ 3 min{𝑚, 𝑛} + 3, 𝑖𝑓 𝑟 ≥ 4

Proof: Since 𝐾

has 𝑚 + 𝑛 vertices and 𝑃

is a path on r vertices, the vertex set of 𝑃

, 𝐾

has r layers (partite sets) of 𝑚 + 𝑛 vertices each. Let 𝑉

, 𝑉

, . . . , 𝑉

denote the r layers and let

𝑎

denotes a vertex in the i

i.e.,𝑉(𝑃

, 𝐾

) = [𝑎

]

𝑟×(𝑚+𝑛)

Let 𝑐(𝑎

) denotes the coloring of the vertex 𝑎

Now, we discuss the star coloring of 𝑃

, 𝐾

in two cases accordingly 𝑚 ≤ 𝑛 𝑎𝑛𝑑 𝑚 > 𝑛.

Case(i): 𝒎 ≤ 𝒏

Here, we discuss the star coloring of 𝑃

, 𝐾

in two subcases:

Subcase(a): 𝟐 ≤ 𝒓 ≤ 𝟑

Now, for 𝑖 = 1, 2, 3 𝑎𝑛𝑑 𝑗 = 𝑚 + 1, 𝑚 + 2, … , 𝑚 + 𝑛, 𝑐(𝑎

) = { 1, 𝑖𝑓 𝑟 = 1,3

2 , 𝑖𝑓 𝑟 = 2 and for 𝑖 = 1, 2, 3 𝑎𝑛𝑑 𝑗 = 1, 2, 3, … , 𝑚,

𝑐(𝑎

) = {

3, 𝑖𝑓 𝑖 + 𝑗 ≡ 2(𝑚𝑜𝑑 𝑚) 4, 𝑖𝑓 𝑖 + 𝑗 ≡ 3(𝑚𝑜𝑑 𝑚)

. . .

𝑚 + 2, 𝑖𝑓 𝑖 + 𝑗 ≡ 0(𝑚𝑜𝑑 𝑚) Subcase(b): 𝒓 ≥ 𝟒

For 𝑖 = 1, 2, 3, … , 𝑟 𝑎𝑛𝑑 𝑗 = 𝑚 + 1, 𝑚 + 2, … , 𝑚 + 𝑛, 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑟 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑟 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑟 ≡ 0(𝑚𝑜𝑑 4) and for 𝑖 = 1, 2, 3, … , 𝑟 𝑎𝑛𝑑 𝑗 = 1, 2, 3, … , 𝑚,

𝑐(𝑎

) = {

4, 𝑖𝑓 𝑖 + 𝑗 ≡ 2(𝑚𝑜𝑑 𝑚) 5, 𝑖𝑓 𝑖 + 𝑗 ≡ 3(𝑚𝑜𝑑 𝑚)

. . .

𝑚 + 3, 𝑖𝑓 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚) Thus, for 𝑚 ≤ 𝑛, the theorem is true.

Case(ii): 𝒎 > 𝒏

Here also, we discuss the star coloring of 𝑃

, 𝐾

in two subcases:

Subcase(a): 𝟐 ≤ 𝒓 ≤ 𝟑

For 𝑖 = 1, 2, 3 𝑎𝑛𝑑 𝑗 = 1, 2, 3, … , 𝑚,

𝑐(𝑎

) = { 1, 𝑖𝑓 𝑟 = 1,3

2, 𝑖𝑓 𝑟 = 2

and for 𝑖 = 1, 2, 3 𝑎𝑛𝑑 𝑗 = 𝑚 + 1, 𝑚 + 2, … , 𝑚 + 𝑛,

𝑐(𝑎

) =

{

3, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 2(𝑚𝑜𝑑 𝑛) 4, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 3(𝑚𝑜𝑑 𝑛)

. . .

𝑛 + 1, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚(𝑚𝑜𝑑 𝑛) 𝑛 + 2, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛).

Subcase(b): 𝒓 ≥ 𝟒

For 𝑖 = 1, 2, 3, … , 𝑟 𝑎𝑛𝑑 𝑗 = 1, 2, 3, … , 𝑚, 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑟 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑟 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑟 ≡ 0(𝑚𝑜𝑑 4) For 𝑖 = 1, 2, 3, … , 𝑟 𝑎𝑛𝑑 𝑗 = 𝑚 + 1, 𝑚 + 2, … , 𝑚 + 𝑛,

𝑐(𝑎𝑖𝑗) = {

4, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 2(𝑚𝑜𝑑 𝑛) 5, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 3(𝑚𝑜𝑑 𝑛)

..

𝑛 + 3, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛).

Thus, we have the star coloring of 𝑃

, 𝐾

from the above cases and it is clear that the star chromatic number of 𝑃

, 𝐾

for given positive integers 𝑚 ≥ 3 and 𝑛, 𝑟 ≥ 2

is

𝜒

(𝑃

, 𝐾

) = { min{𝑚, 𝑛} + 2, 𝑖𝑓 2 ≤ 𝑟 ≤ 3

min{𝑚, 𝑛} + 3, 𝑖𝑓 𝑟 ≥ 4 □

Ilustration 2.1: The star coloring of 𝑃

, 𝐾

is given below:

Fig.:2.1 Star coloring of 𝑷𝟓, 𝑲𝟑,𝟒

n 2.1: The star coloring of 𝑃₅□𝐾_3,4 is given below:

4 5 6 1 1 1 1

(u1, v1) (u1, v2)

5

(u1, v3)

(u1, v4) (u1, v5) (u1, v6) (u1, v7)

(u2, v

5

1)

6 4 2

2

2 2

(u2, v2) (u2, v3) (u2, v4) (u2, v6) (u2, v7)

(u3, v1

6

) (u3, v2

4

) 5

1 1 1 1

(u3, v₃) (u3, v₄) (u3, v₅) (u3, v₆) (u3, v₇)

4 5 6 3 3 3 3

(u4, v₁) (u4, v₂) (u4, v₃)

(u4, v₄) (u4, v₅) (u4, v₆) (u4, v₇)

5 6 4 1 1 1 1

(u5, v1) (u5, v2) (u5, v3)

(u5, v4) (u5, v5) (u5, v6) (u5, v7)

Fig.:2.1 Star coloring of 𝑃₅□𝐾_3,4

Thus, 𝜒

(𝑃

, 𝐾

) = 6.

Corollary 2.2. If 𝑚 = 2 and for any positive integer 𝑛 ≥ 4 and 𝑟 ≥ 3 𝜒

( 𝑃

, 𝐾

) = 5.

Proof: The star coloring pattern for 𝑃

, 𝐾

is as follows For 𝑖 = 1, 2, 3, … , 𝑟 and 𝑗 = 1, 2, 3, … , 𝑚,

𝑐(𝑎

) = {

1, 𝑖𝑓 𝑟 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑟 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑟 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑟 ≡ 0(𝑚𝑜𝑑 4) The star coloring of 𝑎

for 𝑖 = 1, 2, 3, … , 𝑟 and 𝑗 = 1,2 is given by

𝑐(𝑎

) = { 4, 𝑖𝑓 𝑖 + 𝑗 ≡ 2(𝑚𝑜𝑑 𝑚) 5, 𝑖𝑓 𝑖 + 𝑗 ≡ 3(𝑚𝑜𝑑 𝑚) except for 𝑖 ≡ 0(𝑚𝑜𝑑 4).

For 𝑖 = 4,8, . . . , 𝑟 and 𝑗 = 1,2,

𝑐(𝑎

) = { 1, 𝑖𝑓 𝑗 = 1, 𝑖 ≡ 0(𝑚𝑜𝑑 4) 2, 𝑖𝑓 𝑗 = 2, 𝑖 ≡ 0(𝑚𝑜𝑑 4)

Thus, 𝜒

( 𝑃

, 𝐾

) = 5. □

2.2 Star Coloring of 𝑪

, 𝑲

In this section, the star coloring of the cartesian product of the cycle C

with the complete bipartite graph 𝐾

for given positive integers 𝑚, 𝑛, 𝑟 ≥ 2 is discussed.

Lemma 2.3. For given positive integers 𝑚 ≥ 3 and 𝑛, 𝑟 ≥ 2, 𝜒

(𝐶

, 𝐾

) = 𝑚 + 3, if 𝑚 ≤ 𝑛, 𝑛 ≥ 2.

Proof: Let

which has r layers of 𝑚 + 𝑛 vertices and 𝑉(𝐶

, 𝐾

) = [𝑎

]

Consider the vertices 𝑎

, 𝑖 = 1, 2, 3, . . , 𝑟, 𝑗 = 𝑚 + 1, 𝑚 + 2, 𝑚 + 3, … , 𝑚 + 𝑛 . Now the star coloring of these vertices is discussed in the following two cases

Case(i) : r is odd

Subcase(i) : 𝑟 ≥ 3 𝑎𝑛𝑑 𝑟 ≡ 1(𝑚𝑜𝑑 4)

𝑐(𝑎

) = { 𝑖 + 𝑗 − 𝑚 + 3, 𝑖𝑓 𝑖 + 𝑗 ≤ 𝑚 + 𝑛 + 1 5,6,7, … , 𝑛 + 3, 𝑖𝑓 𝑖 + 𝑗 > 𝑚 + 𝑛 + 1

If

for some positive integer 𝑘 ≥ 2, the star coloring of 𝑎

is as follows:

𝑐(𝑎

) = {

3, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) 𝑎𝑛𝑑 𝑖 = 1 4, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) 𝑎𝑛𝑑 𝑖 = 2,3,4, … , 𝑟 − 2

2 , 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) 𝑎𝑛𝑑 𝑖 = 𝑟 − 1, 𝑟

Now, to color 𝑎

if, 𝑖 + 𝑗 = 𝑚 + 𝑘𝑛 + 1 , we use a suitable color of the first m vertices

which are already colored.

We color 𝑎

in the above pattern except for 𝑗 = 𝑟, 𝑖 = 𝑚 + 1, 𝑚 + 2, . . . , 𝑚 + 𝑛 and we color these 𝑎

with the colors used to color 𝑗 = 𝑟 − 2, 𝑖 = 𝑚 + 1, 𝑚 + 2, . . . , 𝑚 + 𝑛.

Subcase(ii) : 𝒓 ≡ 𝟑(𝒎𝒐𝒅 𝟒)

In this case, 𝑐(𝑎

) = 𝑖 + 𝑗 − 𝑚 + 2 𝑖𝑓 𝑖 + 𝑗 ≥ 𝑚 + 𝑛 + 1.

If 𝑖 + 𝑗 = 𝑚 + 𝑘𝑛 + 2, 𝑚 + 𝑘𝑛 + 3, . . . , 𝑚 + 𝑘𝑛 + 𝑟 where 𝑘 ≥ 2 then 𝑐(𝑎

) = 4, 5, 6, . . . , 𝑚𝑖𝑛{𝑚, 𝑛} + 3 respectively.

Case(ii) : r is even

When 𝑟 ≡ 0(𝑚𝑜𝑑 4) , we follow the same pattern as in subcase (ii).

When 𝑟 ≢ 0(𝑚𝑜𝑑 4), 𝑎

is colored with the following pattern:

If 𝑖 + 𝑗 is 𝑚 + 𝑛 + 1, 𝑚 + 𝑛 + 2, 𝑚 + 𝑛 + 3, . . . , 𝑚 + 𝑛 + 𝑟 , we give color 5,6,7,8, . . . , 𝑛 + 3 respectively except for 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) . To color 𝑎

, if 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) , the coloring pattern is as follows:

𝑐(𝑎

) = {

3, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) 𝑎𝑛𝑑 𝑖 = 1 4, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) 𝑎𝑛𝑑 𝑖 = 2,3,4, … , 𝑟 − 2

2, 𝑖𝑓 𝑖 + 𝑗 ≡ 𝑚 + 1(𝑚𝑜𝑑 𝑛) 𝑎𝑛𝑑 𝑖 = 𝑟 − 1, 𝑟

The star coloring of 𝑎

for 𝑖 = 1, 2, 3, . . . , 𝑟 and 𝑗 = 1, 2, 3, . . . , 𝑚 is discussed in the following two cases:

Case(i) : r is odd

If, 𝑟 ≥ 3 𝑎𝑛𝑑 𝑟 ≡ 1(𝑚𝑜𝑑 4) then

𝑐(𝑎

) = {

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

4, 𝑖𝑓 𝑖 = 𝑟 and if, 𝑟 ≥ 3 𝑎𝑛𝑑 𝑟 ≡ 3(𝑚𝑜𝑑 4) then

𝑐(𝑎_𝑖𝑗) = {

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

Subcase(i):

If, 𝑟 = 3 then 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑖 = 1 2, 𝑖𝑓 𝑖 = 2 3, 𝑖𝑓 𝑖 = 3 Case(ii): If r is even

When 𝑟 ≡ 0 (𝑚𝑜𝑑 4), then

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

When 𝑟 ≢ 0 (𝑚𝑜𝑑 4), then 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

4, 𝑖𝑓 𝑖 = 𝑟

Thus, 𝜒

(𝐶

, 𝐾

) = 𝑚 + 3, if 𝑚 ≤ 𝑛, 𝑛 ≥ 2. □ Lemma 2.4. For given positive integers 𝑚 ≥ 3 and 𝑛, 𝑟 ≥ 2, 𝜒

(𝐶

,𝐾

) = 𝑛 + 3, 𝑖𝑓 𝑚 > 𝑛, 𝑛 ≥ 3.

Proof: Let

which has r layers of 𝑚 + 𝑛 vertices and 𝑉(𝐶

, 𝐾

) = [𝑎

]

𝑟×(𝑚+𝑛)

. Consider the vertices of 𝑎

, 𝑖 = 1, 2, 3, . . , 𝑟, 𝑗 = 1, 2, 3, … , 𝑚. Now the star coloring of these vertices is discussed in the following two cases

Case(i): r is odd and 𝑟 ≥ 3 When 𝑟 ≡ 1(𝑚𝑜𝑑 4)

If, 𝑖 + 𝑗 < 𝑛, then 𝑐(𝑎

) = 𝑖 + 𝑗 + 3 except for 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚) .

If, 𝑖 + 𝑗 > 𝑛, i.e., 𝑖 + 𝑗 = 𝑛 + 1, 𝑛 + 2, … , 𝑛 + 𝑟 − 1 then 𝑐(𝑎

) = 5, 6, 7, … , 𝑚 + 3 respectively except for 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚) .

To color 𝑎

𝑐(𝑎

) = {

3, 𝑖𝑓 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)𝑎𝑛𝑑 𝑖 = 1 4, 𝑖𝑓 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)𝑎𝑛𝑑 𝑖 = 2,3, … , 𝑟 − 2

2, 𝑖𝑓 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)𝑎𝑛𝑑 𝑖 = 𝑟 − 1, 𝑟 When 𝑟 ≡ 3(𝑚𝑜𝑑 4),

𝑐(𝑎

) = { 𝑖 + 𝑗 + 2, 𝑖𝑓 𝑖 + 𝑗 < 𝑛

4,5,6, … , 𝑚 + 3, 𝑖𝑓 𝑖 + 𝑗 ≥ 𝑛 𝑎𝑛𝑑 𝑖 + 𝑗 = 𝑘𝑚 + 2, 𝑘𝑚 + 3, … , 𝑘𝑚 + 𝑛 − 1 Case(ii): r is even

When 𝑟 ≡ 0(𝑚𝑜𝑑 4),

𝑐(𝑎

) = { 𝑖 + 𝑗 + 2, 𝑖𝑓 𝑖 + 𝑗 < 𝑛

4,5,6, … , 𝑚 + 3, 𝑖𝑓 𝑖 + 𝑗 ≥ 𝑛 𝑎𝑛𝑑 𝑖 + 𝑗 = 𝑘𝑚 + 2, 𝑘𝑚 + 3, … , 𝑘𝑚 + 𝑛 − 1 When 𝑟 ≢ 0(𝑚𝑜𝑑 4),

𝑐(𝑎_𝑖𝑗) = { 𝑖 + 𝑗 + 3, 𝑖𝑓 𝑖 + 𝑗 < 𝑛 𝑒𝑥𝑐𝑒𝑝𝑡 𝑓𝑜𝑟 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)

5,6, … , 𝑚 + 3, 𝑖𝑓 𝑖 + 𝑗 > 𝑛 𝑎𝑛𝑑 𝑓𝑜𝑟 𝑖 + 𝑗 = 𝑛 + 1, 𝑛 + 2, 𝑛 + 3, … , 𝑛 + 𝑟 − 1 𝑒𝑥𝑐𝑒𝑝𝑡 𝑓𝑜𝑟 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)

To color 𝑎

, if 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚),

3, 𝑖𝑓 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)𝑎𝑛𝑑 𝑖 = 1 4, 𝑖𝑓 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)𝑎𝑛𝑑 𝑖 = 2,3, … , 𝑟 − 2

2, 𝑖𝑓 𝑖 + 𝑗 ≡ 1(𝑚𝑜𝑑 𝑚)𝑎𝑛𝑑 𝑖 = 𝑟 − 1, 𝑟

The star coloring of 𝑎

, for 𝑖 = 1, 2, 3, . . . , 𝑟 and 𝑗 = 𝑚 + 1, 𝑚 + 2, . . . , 𝑚 + 𝑛 is discussed in the following two cases:

Case(i): If r is odd

If

then

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

4, 𝑖𝑓 𝑖 = 𝑟

and if, 𝑟 ≥ 3 𝑎𝑛𝑑 𝑟 ≡ 3(𝑚𝑜𝑑 4) then, 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4)

3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

Subcase(i):

If 𝑟 = 3 , 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑖 = 1 2, 𝑖𝑓 𝑖 = 2 3, 𝑖𝑓 𝑖 = 3 Case(ii): If r is even

When 𝑟 ≡ 0(𝑚𝑜𝑑 4), 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

When 𝑟 ≢ 0(𝑚𝑜𝑑 4), 𝑐(𝑎

) = {

1, 𝑖𝑓 𝑖 𝑖𝑠 𝑜𝑑𝑑

2, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≢ 0(𝑚𝑜𝑑 4) 3, 𝑖𝑓 𝑖 𝑖𝑠 𝑒𝑣𝑒𝑛 𝑎𝑛𝑑 𝑖 ≡ 0(𝑚𝑜𝑑 4)

4, 𝑖𝑓 𝑖 = 𝑟

Thus, 𝜒

(𝐶

,𝐾

) = 𝑛 + 3, 𝑖𝑓 𝑚 > 𝑛, 𝑛 ≥ 3. □ Theorem 2.5. For given positive integers 𝑚 ≥ 3 and 𝑛, 𝑟 ≥ 2,

𝜒

(𝐶

, 𝐾

) = { 𝑚 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 2 𝑛 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 3 Proof: The proof follows from Lemma 2.3 & 2.4

The star chromatic number is,

𝜒

(𝐶

, 𝐾

) = { 𝑚 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 2

𝑛 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 3 □

CONCLUSION

In this paper, it is proved that for given positive integers m ≥ 3 and n, r ≥ 2, (𝑖)𝜒

(𝑃

, 𝐾

) = { min{𝑚, 𝑛} + 2, 𝑖𝑓 2 ≤ 𝑟 ≤ 3

min{𝑚, 𝑛} + 3, 𝑖𝑓 𝑟 ≥ 4 (ii) 𝜒

(𝐶

, 𝐾

) = { 𝑚 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 2

𝑛 + 3, 𝑖𝑓 𝑚 ≤ 𝑛, 𝑛 ≥ 3 REFERENCES

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