CHAPTER 2

A BRIEF LITERATURE REVIEW ON

GRAPH COLORINGS

2.1 THE ORIGIN OF CHROMATIC GRAPH THEORY

The origin of chromatic graph theory can be traced back to 1852 when

Augustus De Morgan wrote a letter to his friend, William Rowan Hamilton, telling

him that one of his students had observed that in coloring a map of England only

four colors were needed in order to ensure that neighboring countries can be assigned

different colors. The four-color conjecture- that every map can be colored with four

colors in this way did not interest Hamilton. This bit of mathematical wisdom was

almost forgotten till the 1860’s when C.S. Peirce presented a proof in a seminar

at Harvard. The conjecture rose to prominence in 1878 when Arthur Cayley, at a

meeting of London Mathematical Society, asked whether or not it had been settled.

Shortly afterwards, Cayley published a short note describing the problem, duly

pointing out where the difficulties lie. A few months later, in the newly founded

American Journal of Mathematics, there appeared what purported to be a proof

of the four- color conjecture. This was A. B. Kempe’s well-known fallacious proof,

which survived for more than ten years before the error was discovered.

The first paper involving edge colorings appeared in 1880. In these pa-

pers P.G.Tait, Professor of Natural philosophy in Edinburgh, out-lined some further

proofs of the four color conjecture, and deduced that edges of every cubic map can

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be colored with just three colors in such a way that the three edges meeting at each

vertex are assigned different colors. He also asserted the converse result without

proof, stating (incorrectly) that one can easily prove by induction that the edges of

every cubic map can be colored with three colors, thereby giving a simple proof of

the four-color conjecture.

In 1890 P.J. Haewood published an important paper refuting Kempe’s proof

and proving that the countries of every map can be properly colored with five colors.

He followed this eight years later with the first of a series of papers in which he took

up Tait’s ideas and reformulated them in terms of congruences. The idea of this

method is to assign one of the numbers +1 or −1 to each vertex of a cubic map in

such a way that the sum of the numbers around each region of the map is congruent

to 0 (mod3). Haewood proved that the four-color conjecture is true if and only if

every set of congruence’s obtained in this way has a non-trivial solution. The next

noteworthy result was due to D. Konig, following the work of Julius Petersen on the

factorization of graphs, Konig proved that if G is any bipartite graph or multigraph

and if the maximum valency of G is , then the edges of G can be colored with

exactly colors in such a way that all of the edges meeting at any vertex are colored

differently.

The results one have described so far involved particular types of graphs,

such as cubic maps or bipartite graphs. The next result is of a more general nature,

and was first proved in an electrical network context by C.E. Shannon in 1949. It

asserts that if G is any graph or multigraph and if the maximum valency of G is ρ ,

then the edges of G can be colored using at most3

2ρ colors. For certain multigraph,

this result is best possible, but for graphs with no multiple edges it can be improved

considerably.

The great breakthrough came in 1964 when V.G. Vizing proved that if G

is any graph with maximum valency ρ , then the edges of G, can always be colored

with only ρ + 1 colors. This magnificent result, generalizes an earlier statement of

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E.I. Johnson that the edges of every cubic graph can be colored with only four colors,

and shows us that every graph with maximum valency ρ can be classified into one

or two classes, according as the number of colors needed is ρ or ρ+ 1. The general

problem of deciding which graphs belong to which class is, however, an exceedingly

difficult one. One of the main methods to attack on this classification problem was

introduced by Vizing in two papers and involves the use of so called critical graphs.

In 1976, the four-color problem, the problem which first gave rise to the

study of edge-colorings, was finally settled using the method of reducible config-

urations and a substantial amount of computer time. K. Appel and W. Haken

successfully proved the four-color conjecture, thereby proving Taits’s result that the

edges of every cubic map can indeed be colored using only three colors.

2.1.1 Applications of Graph Colorings

The problem of coloring a graph has found a number of applications such

as scheduling, register allocation in compilers.

2.1.2 Scheduling of Final Exams

The scheduling problem can be solved using a graph model, with vertices

representing courses and with an edge between two vertices if there is a common

student in the courses they represent. Each time slot for a final exam is represented

by a different color. A scheduling of the exams corresponds to a coloring of the

associated graph.

For example suppose there are seven finals to be scheduled. Suppose the

courses are numbered as 1 to 7. Suppose that the following pairs of courses have

common students.

1 and 2, 1 and 3, 1 and 4, 1 and 7.

2 and 3, 2 and 4, 2 and 5, 2 and 7.

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3 and 4, 3 and 6, 3 and 7.

4 and 5, 4 and 6, 5 and 6, 5 and 7, 6 and 7.

Figure 2.1: Graph model showing scheduling.

Since the chromatic number of the above graph is 4, four time slots are

needed. One can arrange them as follows:

Time Period Courses

I 1-6

II 2

III 3-5

IV 4-7

2.1.3 Index Registers

In efficient compilers the execution of loops is speeded up when frequently

used variables are stored temporarily in index registers in the central processing

unit, instead of in regular memory. For a given loop, how many index registers are

needed? This problem can be addressed using a graph coloring model. To setup the

model, let each vertex of a graph represent a variable in the loop. There is an edge

between two vertices if the variable they represent must be stored in index registers

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at the same time during the execution of the loop. Thus the chromatic number of

the graph gives the number of index registers needed, since different registers must

be assigned to variables when the vertices representing these variables are adjacent

in the graph.

2.2 BASIC DEFINITION

2.2.1 Proper Vertex Coloring

An assignment of colors to the vertices of a graph so that no two adjacent

vertices have the same color.

2.2.2 Chromatic Number

The least number of colors needed to color the graph is called its chromatic

number and it is denoted by χ . A graph that can be assigned a proper k-coloring

is k colorable, and it is k-chromatic if its chromatic number is exactly k.

2.2.3 Some Properties of χ(G):

• χ(G) = 1 if and only if is totally disconnected.

• χ(G) ≥ 3 if and only if G has an odd cycle. Equivalently, if G is not bipartite.

• χ(G) ≤ ∆(G) + 1.

• Brooks theorem

χ(Kn) = χ(C2n+1) = ∆ + 1 (n ∈ N)

χ(G) ≤ ∆ in any other case (n ∈ N)

• χ(G) ≤ 4 for any planar graph G (Four-color theorem).

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2.2.4 Independent Vertex

A set of vertices in G is independent if no two of them are adjacent. The

largest number of vertices in a set is called the vertex independence number of G

and is denoted by β0(G) or β0 .

2.2.5 Color Class

The set of all vertices with any one color is independent and is called a

color class.

2.2.6 Decision Problem

A decision problem is a language L of strings over an alphabet.

2.2.7 P and NP

P is the class of decision problems for which one can find a solution in

polynomial time. NP is the class of decision problems for which one can check

solutions in polynomial time. NP stands for Non deterministic Polynomial time.

2.2.8 NP-Hard

A decision problem L is NP-hard if for every language L′ in NP, L′ is

polynomial time reducible to L. (A language L′ polynomial time reducible to a

language L if there exist a polynomial time function f(x) from string to string such

that x is in L′ if f(x) is in L.)

2.2.9 NP-Complete

A decision problem L is NP-complete if it is in NP and it is NP-hard.

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2.3 OTHER TYPES OF COLORINGS

Definition 2.3.1. Let G be a graph. A vertex coloring of G assigns colors,

usually denoted by 1, 2, 3, . . . , n to the vertices of G. One color per vertex, so that

adjacent vertices are assigned different colors.

Definition 2.3.2. A k-coloring of G is a coloring which consists of k different colors

and in this case G is said to be k-colorable.

Definition 2.3.3. A map is defined to a be a plane connected graph with no bridges.

A map G is said to be k-face colorable if one can color its faces with at most k

colors in such a way that no two adjacent faces i.e., two faces sharing a common

boundary edge, have the same color.

Definition 2.3.4. An edge coloring of graph G = (V,E) is a function C : E → N ,

in which any two adjacent edges e, f ∈ E are assigned different colors. The function

C is known as the edge-coloring function. A graph G for which there exists an

edge-coloring which requires k colors is called k-edge colorable, while such a coloring

is called a k-edge coloring. The smallest number k of which there exists a k - edge

- coloring of G is called the chromatic index of a graph G and is denoted by χ′(G).

Any edge - coloring of G which requires k = χ′(G) colors is called chromatic or

optimal.

Definition 2.3.5. A co-coloring of a graph G is an assignment of colors to the

vertices such that each color class forms an independent set in G or the complement

of G. The co-chromatic number z(G) of G is the least number of colors needed in

any co-colorings of G.

Definition 2.3.6. A grundy coloring of a graph is a proper vertex coloring φ :

V → N such that every vertex v has a neighbor of color i for all 1 ≤ i ≤ φ(v).

Definition 2.3.7. An exact coloring is a (proper) vertex coloring in which every

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pair of colors appears on exactly one pair of adjacent vertices. In essence, an exact

coloring is a coloring that is both harmonious and complete.

Definition 2.3.8. A harmonious coloring is a proper vertex coloring in which

every pair of colors appears on at most one pair of adjacent vertices. The harmonious

chromatic number χH(G) of a graph G is the minimum number of colors needed for

any harmonious coloring of G.

Definition 2.3.9. A complete coloring is the opposite of harmonious coloring in

the sense that it is a vertex coloring in which every pair of colors appears on at least

one pair of adjacent vertices. The achromatic number ψ(G) of a graph G is the

maximum number of colors needed for any complete coloring of G.

Definition 2.3.10. A star coloring of a graph G is a proper vertex coloring in

which every path on four vertices uses at least three distinct colors. Equivalently, in

a star coloring, the induced subgraphs formed by the vertices of any two colors has

connected components that are star graphs. The star chromatic number χs (G) of G

is the least number of colors needed to star color G.

2.4 EQUITABLE COLORING

Definition 2.4.1. If the set of vertices of a graph G can be partitioned into k classes

V1, V2, . . . , Vk such that each Vi is an independent set and the condition ||Vi|−|Vj|| ≤

1 holds for every i 6= j, then G is said to be equitably k-colorable (V1, V2, . . . , Vk)

is called an equitable independence-partition. The smallest integer k for which

G is equitable k− colorable is known as the equitable chromatic number [2, 3,

6, 8, 9, 11, 12, 19, 20, 21, 27, 35, 42, 43, 47, 48] of G and denoted by χ=(G).

Definition 2.4.2. [46] The equitable chromatic threshold of G is

χ∗

= = min {t : G is equitable k-colorable for all k ≥ t}.

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Definition 2.4.3. Let G = (V (G), E(G)) be a graph with vertex set V and edge set

E. A k-total coloring of a graph G is a coloring of the vertices and edges of G with

k colors so that no two adjacent or incident elements have the same color. The total

chromatic number χ′′(G) of G is the smallest k such that G has a k-total coloring.

A total coloring is equitable [18, 26, 28, 36, 51, 53] if the number of vertices

and edges colored with each color differs by at most one. The smallest k for which

G has such a coloring is named equitable total chromatic number and denoted

by χ′′

=(G).

In some discrete industrial systems one can encounter the problem of eq-

uitable partitioning of a system with binary conflicting relations into conflict-free

sub-systems. Such situations can be modeled by means of equitable graph coloring.

For example, in garbage collection problem [52] the vertices of the graph

represent garbage collection routes and pair of vertices is joined by an edge if the

corresponding routes should not be run on the same day the problem of assigning

one of the six days of the work week to each route thus reduces to the problem of

6− coloring of the graph. In practice it might be desirable to have an approximately

equal number of routes run on each of the six days. So one have to color the graph

in an equitable way with six colors.

Another application of equitable edge-coloring can be found in scheduling

the timetabling. For example, if we want to create university timetables, such that

every class and every teacher has the same number of lectures every days, one have

to construct a bipartite graph G(X, Y ), where X is the set of vertices corresponding

to teachers, and Y is the set of vertices corresponding to classes. Two vertices

x ∈ X and y ∈ Y are joined by an edge if and only if the class y has lectures with

teacher x. The problem of finding “an equitable” timetable reduces to an equitable

edge-coloring of G(X, Y ).

One can observe certain obvious results (i) χ=(Kn) = n, (ii) χ=(Kn,n) = 2,

(iii) χ=(Pn) = 2 and hence (iv) χ= (L(Pn)) = 2.

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2.5 LITERATURE SURVEY

Literature survey is the background work that is done personally. It is

based on books and academic publications. The motive of literature survey is to

gather a basis for the partial work and the researcher will be familiar with existing

literature and research on the topic. It is not smart to re-invent the wheel, i.e., one

has to know what has already been done. The list of literature survey on equitable

coloring can be present as follows.

This notation of equitable colorability was first introduced in a paper by

Walter Meyer [48]. His motivation came from Tucker [52] where vertices representd

garbage collecstion routes and two such vertices were joined when the corresponding

routes should not be run on the same day. Meyer thought that it would be desirable

to have an approximately equal number of routes run on each of the six days.

The main result obtained by Meyer was that a tree T can be equitably

colored with ⌈∆ (T ) /2⌉+ 1 colors. Unfortunately, his proof was marred with gaps.

According to Guy’s report [24], Eggleton could extend Meyer’s result to show that

a tree T can be equitably colored with k colors, provided k ≥ ⌈∆ (T ) /2⌉ + 1. A

finer result about trees is the following theorem by Bollobas and Guy [3]. However

a complete determination of when a tree is equitably colorable was left unsolved in

their paper.

Theorem 2.5.1. [3] A tree T on t vertices is equitably 3-colorable if t ≥ 3∆ (T )−8

or t = 3∆ (T )− 10.

By far the most interesting contribution made in meyer’s paper is to propose

the following conjecture.

The Equitable Coloring Conjecture (ECC) Let G be a connected

graph. If G is neither a complete graph nor an odd cycle, then χ= (G) ≤ ∆ (G)

Probably due to the lack of using powerful computers, Meyer was successful

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in verfying the ECC only for graphs with six or fewer vertices. Apparently the

motivation of the ECC came from the following fundamental theorem of Brooks [5].

Theorem 2.5.2. [5] Let G be a connected graph. If G is neither a complete graph

nor an odd cycle, then χ (G) ≤ ∆ (G)

One well-known result of Hajnal and Szemeredi [25], when rephrased in

terms of the equitable colorability, has already shown the following.

Theorem 2.5.3. [25] A graph G (not necessarily connected) is equitable k-colorable

if k ≥ ∆ (G) + 1.

In an entirely different context, de Werra [57] treated color sequences and

the majorization ordering among them. However his results have consequence in

the equitable colorability. A sequence of nonnegative integers h = (h1, h2, · · · , hk)

is called a color sequence for a given graph G if the following conditions hold.

• h1 ≥ h2 ≥ · · · ≥ hk ≥ 0;

• There is a k-coloring of G such that the color classes V1, V2, · · · , Vk satisfy

|Vi| = hi for i = 1, 2, · · · , k.

Let Km,n denote the complete bipartite graph whose parts are of size m and n,

respectively. A claw-free graph is a graph containing noK1,3 as an induced subgraph.

One of de Werra’s result is the following.

Theorem 2.5.4. [57] Let G be a claw-free graph and h = (h1, h2, · · · , hk) be color

sequence of G. Then any sequence of nonnegative integers h′ = (h′1, h′

2, · · · , h′

k) is

also a color sequence if h′ is majorized by h.

Theorem 2.5.2 and Theorem 2.5.4, it follows that, for a claw-free graph G, G

is equitably k-colorable for all k ≥ χ (G), or equivalently χ∗

= (G) = χ= (G) = χ (G).

Although this fact can be shown directly, it was first implicitly implied in de Werra’s

paper. One can immediately see that the ECC holds for claw-free graphs. Since

every line graph is claw-free, the ECC holds for line graphs in particular.

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2.5.1 Bipatite Graphs

A breakthrough about the equitable coloring was achived when Lih and

Wu [44] settled the ECC for bipartite graphs.

Theorem 2.5.5. [44] Let G = G (X,Y ) be a connected bipartie graph. If G is

different from any complete bipartite graph Kn,n, then G can be equitably colored

with ∆ (G) colors.

Theorem 2.5.6. [44] The complete bipartite graph Kn,n can be equitably colored

with k colors if and only if ⌈n/ ⌊k/2⌋⌉ − ⌊n/ ⌈k/2⌉⌋ ≤ 1.

In many cases the equitable chromatic number is far below the maximum

degree. If one can impose additional constraints on the graph, one could arrrive at

a better bound for the equitable chromatic number. The following is a result of this

type.

Theorem 2.5.7. [44] Let G = G (X, Y ) be a connected bipartite graph with ǫ edges.

Suppose |X| = m ≥ n = |Y | and ǫ < ⌊m/ (n+ 1)⌋ (m− n) + 2m. Then χ= (G) ≤

⌈m/ (n+ 1)⌉+ 1

The bound for the equitable chromatic number in the above theorem is

indeed better than ∆ (G) when there are at least two edges. It is also trivial to see

that the conjecture holds for complete bipartite graphs. The Meyer-Eggleton result

about trees gives another evidence for the conjecture.

Conjecture 2.5.8. Let G be a connected bipartite graph.

Then χ= (G) ≤ ⌊(∆ (G) + 3) /2⌋.

There are some miscellaneous results on the equitable coloring of general

bipartite graphs. For instance, Lin in his Master degree thesis [45] proved the

following.

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Theorem 2.5.9. [45] Let G (A,B) be a bipartite graph. Let |G| abbreviate |V (G)|

and ∆G (B) denote the maximum degree of a vertex in part B. If |B| < ⌊|G| /k⌋

and ∆G (B) ≤ r (k − r), where k − ⌊|G| /k⌋ ≤ r ≤ k, then χ∗

= (G) ≤ k.

Theorem 2.5.10. [45] Let G (A,B) be a bipartite graph. then χ∗

= (G) ≤ 2k − 2 if

one of the following two conditions is true.

1. |G| ≥ k2 and ∆G (B) ≤ ⌈k/2⌉ ⌊k/2⌋, or

2. |G| < k2 and ∆G (B) ≤ ⌈|G| /2k⌉ ⌊k − |G| /2k⌋.

2.5.2 Trees

A graph is non-trivial if it contains at least one edge. There is natural

way to regard a non-trivial tree T as a bipartite graph T (X, Y ). The technique

one can use to prove the ECC for connected bipartite graphs can be applied to

find the equitable chromatic number of a non-trivial tree when the two parts differ

in sizes by at most one. Try to cut the parts into classes of nearly equal size. If

there are vertices remaining, then manage to find the non adjacent vertices in the

opposite part to form a class of the right size. In Chen and Lih [8], the following

was established.

Theorem 2.5.11. [8] Let T = T (X,Y ) be a non-trivial tree satisfying ||X| − |Y || ≤

1. Then χ= (T ) = χ∗

= (T ) = 2

Theorem 2.5.12. [8] Let T = T (X,Y ) be a tree satisfying ||X| − |Y || > 1. Then

χ= (T ) = χ∗

= (T ) = max {3, ⌈(|T |+ 1) / (α (T −N (v)) + 1)⌉}, where v is an arbi-

trary major vertex.

2.5.3 The Equitable ∆-Coloring Conjecture

In contrast to the ordinary proper coloring of a graph, the equitable col-

oring does not posses monotonicity, namely, a graph could be equitable k-colorable

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without being equitably (k + 1)-colorable. Therefore the ECC does not fully reveal

the nature of the equitable coloring. It seems that the maximum degree plays a

crucial role here. After examining relevent evidence, Chen, Lih and [9] proposed the

following.

The Equitable ∆-Coloring Conjecture (E∆CC) Let G be a connected

graph. If G is not a complete graph, or an odd cycle, or a complete bipartite graph

K2m+1,2m+1, then G is equitably ∆ (G)-colorable.

The conclusion of the E∆CC can be equivalently stated as χ∗

= (G) ≤ ∆ (G).

It is also immediate to see that the E∆CC implies the ECC. On the reverse direction,

if the ECC is true, so is the E∆CC for non regular graphs. In Chen, Lih and Wu

[7], the E∆CC was settled for graphs whose maximum degree is at least one-half

of the order. The following two lemmas supplied the basic tools for the solution.

Use G to denote the complement graph of G. δ (G) and α′ (G) denote the minimum

degree and the edge-independence number of G, respectively.

Theorem 2.5.13. [7] Let G be a disconnected graph. If G is different from (Km)

and (K2m+1,2m+1) for all m ≥ 1, then α′ (G) > δ (G).

Theorem 2.5.14. [7] Let G be a connected graph such that |G| > dδ (G) + 1.

Suppose the vertex set of G cannot be partitioned into a set H of size δ (G) and an

independent set I of size |G| − δ (G) such that each vertex of I is adjacent to all

vertices of H. Then α′ (G) > δ (G).

Theorem 2.5.15. Let G be a connected graph with ∆ (G) ≥ |G| /2. If G is different

from Km and K2m+1,2m+1 for all m ≥ 1, then G is equitably ∆ (G)-colorable.

As pointed out by Yap, a close examination of the proof of Theorem 2.5.15

in Chen, Lih and Wu[9] reveals that a stronger result was obtained, namely, χ∗

= ≤

|G| − α′(G) ≤ ∆ (G). In a similar vein, Yap and Zhang [59] made an analy-

sis of the complement graph and succeeded in extending the above result. Let r

be the maximum number of vertex-disjoint tringles in G, s the maximum number

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of independent edges in G −r⋃

i=1

{ai, bi, ci} for all possible vertex-disjoint triangles

{a1, b1, c1} , · · · {ar, br, cr} in G, and t = |G| − 3r − 2s. Yap and Zhang proved the

following.

Theorem 2.5.16. Let G be a connected graph such that (|G| /3) + 1 ≤ ∆ (G) <

|G| /2. Then χ∗

= (G) ≤ r + s+ t ≤ ∆ (G).

Theorem 2.5.15 and Theorem 2.5.16 together establish the E∆CC for the

special classes of graphs. By attaching appropriately auxiliary graphs to a non-

regular graph, let us restrict our attention to regular graphs. An equitable coloring

of the extended regular will induce an equitable coloring to the original non-regular

graph. Chen, Lih and Wu [9] followed this strategy to show the next reduction

result.

Theorem 2.5.17. The E∆CC holds if it does so for all regular graphs.

The first non-trivial case to investigate is cubic graphs, i.e., regular graphs

of degree 3. If the chromatic number of a connected cubic graph G is 2, then the

E∆CC has already been established. The following theorem shows the equitqble

coloring on cubic graphs.

Theorem 2.5.18. Let G be a connected cubic graph whose chromatic number is 3.

Then χ= (G) = 3.

Theorem 2.5.19. Let G (X, Y ) be a connected bipartite graph such that |X| = m ≥

n = |Y | and ∆ (G (X,Y )) ≤ 3. If |Y 3X| = t, Then m− n ≤ t+ 1.

2.5.4 Special Families Supporting E∆CC

There are interesting results dealing with special families of graphs that

provided positive evidence for the E∆CC. Let us start with the family of split graphs.

A connected graph G is called a split graph if its vertex set can be partitioned into

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two nonempty subsets U = {u1, u2, . . . , un} and V = {v1, v2, . . . , vr} such that U

induces a complete graph and V induces an independent set. Denote the split graph

G as G[U ;V ] and always assume that no vertex in V is adjacent to all vertices in U .

Assign a family of bipartite graphs BG(k), k ≥ 1, to the given split graph G[U ;V ] in

the following way. The vertex set of BG (k) is {uij : 1 ≤ i ≤ n and 1 ≤ j ≤ k}⋃

V

and {uij, vt} is defined to be an edge of BG(k) if and only if ui and vt are non-

adjacent in G. Note that BG(k) is a subgraph of BG(k+1). The coloring of a split

graph G[U ;V ] is closely related to independent edges of the graphs BG(k). For

instance, any given set of independent edges in BG(k) induces a partial coloring of

G in the following standard way. Use the ith color to color ui and all those vertices

in V that are matched by the edges to some uij, 1 ≤ j ≤ k. Chen, Ko and Lih [6]

proved the following.

Theorem 2.5.20. Let G [U ;V ] be a split graph such that |U | = n and |V | = r. Let

m = max {k : α′ (BG (k)) = kn} if the set in question is nonempty; otherwise let m

be zero. Then χ∗

= (G [U ;V ]) = n+ ⌈(r − α′ (m+ 1)) / (m+ 2)⌉.

Once χ∗

= (G [U ;V ]) is know, it is straightforward to verify the E∆CC for

split graphs.

A graph is planar if it can be drawn on Euclidean plane such that edges only

meet each other at points reprecenting the vertices of the graph. An Outerplanar

graph is a planar graph that has a drawing on the plane such that every vertex lies

on the unbounded face. Zhang and Yap[61] proved the following.

Theorem 2.5.21. A planar graph G is equitably ∆ (G)-colorable if ∆ (G) ≥ 13

In an earlier paper, Yap and Zhang [60] gave an elegant proof to settle the

case for outerplanar graphs.

Theorem 2.5.22. Let G be an outerplanar graph having ∆ (G) ≥ 3, then G is

equitably ∆ (G)-colorable.

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2.6 MISCELLANEOUS RESULTS

Let Kn1,n2,...,ntdenote the complete t−partite graph whose parts have sizes

n1, n2, · · · , nt. Chen and Wu [12] proved the following two theorems.

Theorem 2.6.1. χ= (Kn1,n2,··· ,nt) =

t∑

i=1

⌈ni/h⌉ where = max {k : ni ≥ ⌈ni/k⌉ (k − 1)

for all i}.

Theorem 2.6.2. χ∗

= (Kn1,n2,··· ,nt) =

t∑

i=1

⌈ni/h⌉ where = min {k : there is i such that

ni < ⌈ni/ (k + 1)⌉ k or there are ni and nj, i 6= j, such that k divides neither ni nor

nj}.

Take two graphs G1 and G2. The cartesian product of the vertex sets

V (G1) × V (G2) to be the vertex set of a new graph. There are several ways to

define the edge set of a product graph. Let us introduce two products [40], the

square product G1�G2 and the cross product G1×G2. They are also known as the

cartesian and the direct products, respectively. The edge sets are defined as follows.

E (G1�G2) = {(u, x) (v, y) : (u = v and xy ∈ E (G2)) or (x = y and uv ∈ E (G1))},

E (G1 ×G2) = {(u, x) (v, y) : uv ∈ E (G1) and xy ∈ E (G2)}.

The following results concerning products are included in a note by Chen,

Lih and Yan[10]

Theorem 2.6.3. χ∗

= (G1) ≤ ∆ (G1)+1 and χ∗

= (G2) ≤ ∆ (G2)+1, then χ∗

= (G1�G2) ≤

max {∆ (G1) + 1,∆ (G2) + 1}. When both G1 and G2 are non-trivial graphs, ∆ (G1�G2) ≥

max {∆ (G1) + 1,∆ (G2) + 1}. Therefore G1�G2 is equitably ∆ (G1�G2)−colorable.

Theorem 2.6.4. (1) χ= (Kn�Km) = χ∗

= (Kn�Km) = max {n,m}.

(2) χ= (Cn�Cm) = χ∗

= (Cn�Cm) =

2 if n,m are even

3 otherwise

.

Theorem 2.6.5. (1) χ= (Kn ×Km) = min {n,m}.

29

(2) χ= (Cn × Cm) = χ∗

= (Cn × Cm) =

2 if nm is even

3 otherwise

.

In the following Chapters, let us discuss about the corona product of various

special families of graphs in detail.