Using latin squares Using latin squares to color split graphs to - - PowerPoint PPT Presentation

Using latin squares Using latin squares to color split graphs to color split graphs

Sheila Morais de Almeida Sheila Morais de Almeida (State University of Campinas)

(State University of Campinas)

C Cé élia Picinin de Mello lia Picinin de Mello (State University of Campinas)

(State University of Campinas)

Aurora Morgana Aurora Morgana (University of Rome

(University of Rome “ “La Sapienza La Sapienza” ”) )

May - 2008

Outline Outline

• Edge

Edge-

• Coloring

Coloring

• The Classification Problem

The Classification Problem

• Split Graphs

Split Graphs

• Overfull

Overfull Graphs Graphs

• The Edge

The Edge-

• Coloring

Coloring Conjecture Conjecture for Split Graphs for Split Graphs

• Latin Squares

Latin Squares

• The Edge

The Edge-

• Coloring of Split Graphs Using Latin

Coloring of Split Graphs Using Latin Squares Squares

Edge Edge-

• Coloring

Coloring

A k k-

• edge

edge-

• coloring

coloring

• f a graph G is an

assignment of k colors to the edges of G such that any two edges incident in a common vertex have distinct colors.

The Edge The Edge-

• Coloring Problem

Coloring Problem

The minimum k required to perform a k-edge-coloring of a simple graph G is called chromatic chromatic index index of G and is denoted by χ’(G).

χ’(Bull)=3

The Edge The Edge-

• Coloring Problem

Coloring Problem

χ χ’

’(G) (G) ≥ ≥ ∆

∆(G),

(G), where ∆(G) is the maximum degree of a graph G.

χ’(Bull)=3

The Edge The Edge-

• Coloring Problem

Coloring Problem

There are graphs that have

χ’(G) > ∆(G).

In 1964, Vizing showed that every simple graph G has

χ’(G) ≤ ∆(G)+1. ∆(C3)=2 χ’(C3) = 3

The Edge The Edge-

• Coloring Problem

Coloring Problem

A direct result of the Vizing’s Theorem is that any simple graph G has

∆(G) ≤ χ’(G) ≤ ∆(G)+1.

Vizing restricted the Edge-Coloring Problem to the following problem: Given a simple graph G, is the chromatic index of G equal to ∆(G) or ∆(G)+1?

The Classification Problem The Classification Problem

Given a simple graph G, is the chromatic index of G equal to ∆(G) or ∆(G)+1? This problem is known as the Classification Classification Problem Problem.

The Classification Problem The Classification Problem

If χ’(G) = ∆(G), then G is Class 1. Otherwise, χ’(G) = ∆(G) + 1 and G is Class 2.

Classe 1 Classe 2

The Classification Problem The Classification Problem

In 1981, Holyer showed that the problem

• f deciding if a simple graph G is Class 1 is

NP-Complete.

The Classification Problem The Classification Problem

Even so, there are efficient algorithms to solve this problem when we are restricted to some classes of graphs.

bipartite graphs are Class 1. a complete graph is Class 1 iff it has an even number

• f vertices.

a cycle (without chords) is Class 1 iff it has an even

number of vertices.

split graphs with odd maximum degree are Class 1.

Split Graphs Split Graphs

A graph G is a split graph split graph if the set of vertices of G admits a partition [Q, S], where Q is a clique and S is a stable set. Q S

Overfull Overfull Graphs Graphs

Consider a graph G, with n vertices and m edges. The graph G is

• verfull
• verfull

if n is

• dd and

m > ∆(G) n/2.

m = 9 > ∆(G) n/2 = = 4 5/2 = 8

Subgraph Subgraph-

• Overfull

Overfull Graphs Graphs

If G has a subgraph which is overfull and has maximum degree equal to ∆(G), then G is subgraph subgraph-

• overfull
• verfull.

Subgraph Subgraph-

• Overfull

Overfull Graphs Graphs

If G has a subgraph which is overfull and has maximum degree equal to ∆(G), then G is subgraph subgraph-

• overfull
• verfull.

m = 9 > ∆(G) n/2 = = 4 5/2 = 8

Neighborhood Neighborhood-

• Overfull

Overfull Graphs Graphs

If G has an overfull subgraph induced by a vertex with degree ∆(G) and its neighbors, then G is neighborhood neighborhood-

• verfull
• verfull.

m = 9 > ∆(G) n/2 = = 4 5/2 = 8

Subgraph Subgraph-

• Overfull

Overfull Graphs Graphs

Overfull Overfull graphs graphs and and neighborhood neighborhood-

• overfull
• verfull

graphs are subgraph graphs are subgraph-

• overfull graphs.
• verfull graphs.

Every Every subgraph subgraph-

• overfull
• verfull graph is

graph is Class Class 2. 2.

Overfull Class 2 Subgraph-overfull Neighborhood-

• verfull

Edge Edge-

• Coloring

Coloring Conjecture Conjecture for Split for Split Graphs Graphs

Figueiredo, Figueiredo, Meidanis Meidanis and and Mello show Mello show that that every every subgraph subgraph-

• overfull
• verfull split graph is

split graph is neighborhood neighborhood-

• overfull
• verfull.

. They They present present the the following following conjecture conjecture: : A split graph G is A split graph G is Class Class 2 if, 2 if, and and only

• nly if, G

if, G is is neighborhood neighborhood-

• overfull
• verfull.

.

Graphs Graphs with with Universal Universal Vertices Vertices

Planthold Planthold presents presents the the following following theorem theorem: : Every Every simple graph G simple graph G containing containing a a universal universal vertex vertex is is Class Class 2 2 iff iff G is G is subgraph subgraph-

• overfull
• verfull.

.

K5 minus one edge is overfull K5 is subgraph-overfull

Split Graphs Split Graphs

In 1995, Chen, Fu, and Ko showed that split graphs with odd ∆(G) are Class 1.

∆(G)=7

Split Graphs Split Graphs

Every split graph G with partition [Q, S] has a bipartite subgraph induced by the edges with a vertex in Q and another vertex in S. Q S

Split Graphs Split Graphs

Considering the bipartite subgraph of G, we denote: d(Q) = max{d(v), v ∈ Q} and d(S) = max{d(v), v ∈ S}.

Q S d(Q) = 2 d(S) = 3

Split Graphs Split Graphs

Consider the partition [Q, S] of a split graph, where Q is a maximal clique. Chen, Fu and Ko also showed that every split graph with d(Q) ≥ d(S) is Class 1.

Q S d(Q) = 3 d(S) = 1

Edge Edge-

• Coloring of Split Graphs

Coloring of Split Graphs

Split Graphs that are neighborhood-overfull are Class 2. Split Graphs with odd maximum degree are Class 1. Split Graphs with even maximum degree that are not

neighborhood-overfull and contain a universal vertex are Class 1.

Split-Graphs with partition [Q, S], where Q is a maximal

clique and such that d(Q) ≥ d(S) are Class 1.

How about split graphs with even maximum degree and d(S) > d(Q), that are not neighborhood-overfull and do not contain universal vertices? Are these graphs Class 1?

Latin Latin Square Square

A A latin latin square square of

• f order
• rder k

k is is

• a

a k kx

xk

k-

• matrix

matrix

• filled

filled with with entries entries from from {0, 1, ..., k {0, 1, ..., k-

• 1}

1}

• each

each element element appears appears exactly exactly once

• nce in

in each each row row

• each

each element element appears appears exactly exactly once

• nce in

in each each column column. .

4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

Commutative Commutative Latin Latin Square Square

A latin A latin square square M=[m M=[mi,j

i,j] is

] is commutative commutative if if m mi,j

i,j =

= m mj

j,i ,i for 0

for 0 ≤ ≤ i,j i,j ≤ ≤ k k-

• 1.

1. 4 3 2 1 4 3 2 1 1 4 3 2 2 1 4 3 3 2 1 4 4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

Commutative Commutative Latin Latin Square Square

A latin A latin square square M=[m M=[mi,j

i,j] is

] is commutative commutative if if m mi,j

i,j =

= m mj

j,i ,i for 0

for 0 ≤ ≤ i,j i,j ≤ ≤ k k-

• 1.

1. 4 4 3 3 2 2 1 1 4 4 3 3 2 2 1 1 1 1 4 4 3 3 2 2 2 2 1 1 4 4 3 3 3 3 2 2 1 1 4 4 4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

4 3 2 1 4 3 2 1 1 4 3 2 2 1 4 3 3 2 1 4 4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

Commutative Commutative Latin Latin Square Square

A latin A latin square square M=[m M=[mi,j

i,j] is

] is commutative commutative if if m mi,j

i,j =

= m mj

j,i ,i for 0

for 0 ≤ ≤ i,j i,j ≤ ≤ k k-

• 1.

1.

4 3 2 1 4 3 2 1 1 4 3 2 2 1 4 3 3 2 1 4 4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

Commutative Commutative Latin Latin Square Square

A latin A latin square square M=[m M=[mi,j

i,j] is

] is commutative commutative if if m mi,j

i,j =

= m mj

j,i ,i for 0

for 0 ≤ ≤ i,j i,j ≤ ≤ k k-

• 1.

1.

Idempotente Latin Idempotente Latin Square Square

A latin A latin square square M=[m M=[mi,j

i,j] is

] is idempotente idempotente if if m mi,i

i,i = i for 0

= i for 0 ≤ ≤ i i ≤ ≤ k k-

• 1.

1. 2 4 1 3 4 1 3 2 1 3 2 4 3 2 4 1 2 4 1 3 4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

Idempotente Latin Idempotente Latin Square Square

A latin A latin square square M=[m M=[mi,j

i,j] is

] is idempotente idempotente if if m mi,i

i,i = i for 0

= i for 0 ≤ ≤ i i ≤ ≤ k k-

• 1.

1. 2 4 1 3 4 1 3 2 1 3 2 4 3 2 4 1 2 4 1 3 4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

Idempotente Latin Idempotente Latin Square Square

A latin A latin square square M=[m M=[mi,j

i,j] is

] is idempotente idempotente if if m mi,i

i,i = i for 0

= i for 0 ≤ ≤ i i ≤ ≤ k k-

• 1.

1. 2 4 1 3 4 1 3 2 1 3 2 4 3 2 4 1 2 4 1 3 4 3 2 1 3 2 1 4 2 1 4 3 1 4 3 2 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

Chen Chen, , Fu Fu and and Ko Ko use use idempotente idempotente commutative commutative latin squares latin squares to to prove prove that that split split graphs graphs with with odd

• dd

maximum maximum degree degree are are Class Class 1. 1. 5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

v0 v1 v2 v3 v4

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

v0 v1 v2 v3 v4 1 2 3 4 0 1 2 3 4

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

v0 v1 v2 v3 v4

2 2 2 2 5 5 5 5 1 1 4 4 6 6 6 6 3 3

1 2 3 4 0 1 2 3 4

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

v0 v1 v2 v3 v4

2 2 2 2 5 5 5 5 1 1 4 4 6 6 6 6 3 3

1 2 3 4 0 1 2 3 4

1 1 2 2

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

v0 v1 v2 v3 v4

2 2 2 2 5 5 5 5 1 1 4 4 6 6 6 6 3 3

1 2 3 4 0 1 2 3 4

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

5 1 4 3 6 2 2 5 1 4 3 6 2 5 1 4 5 3 6 2 3 5 1 4 6 1 4 3 2 4 3 6 1 3 6 2 5 4 6 2 5 1

v0 v1 v2 v3 v4 1 2 3 4 0 1 2 3 4

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

Let Let d di

i be

be the the number number of

• f elements

elements in in an an array array C Ci

i.

. Let Let c ci

i,j, ,j, be

be the the j jth

th entrie

entrie of the

• f the

array array C Ci

i. .

A set of A set of arrays arrays {C {C0

0, ...,

, ..., C Ck

k} is a

} is a monotonic monotonic color color diagram diagram if if c ci

i,j, ,j, occurs

• ccurs at

at most most d di

i-

• j times in

j times in the the arrays arrays {C {C0

0, C

, C1

1, ...,

, ..., C Ci

i-

• 1

1} .

} . 5 1 4 1 4 3 4 3 3 6

C0 C1 C2 C3 C4 1 2 3

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

It is It is always always possible possible to color a to color a bipartite bipartite graph graph [Q,S] [Q,S] with with a a monotonic monotonic color color diagram diagram {C {C0

0, C

, C1

1, ...

, ... C C|Q

|Q| |} if

} if C Ci

i has

has size size at at least least d(v d(vi

i).

). 5 1 4 1 4 3 4 3 3 6

C0 C1 C2 C3 C4 0 1 2 v0 v1 v2 v3 v4

4 4 4 4 3 3 4 4 1 3 3

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

v0 v1 v2 v3 v4

2 2 2 2 5 5 5 5 1 1 4 4 6 6 6 6 3 3 4 4 4 4 3 3 4 4 1 3 3 1 1 2 2

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

It is It is not not possible possible to to construct construct an an idempotente idempotente commutative commutative latin latin square square of

• f even

even order

• rder.

. 3 1 4 5 2 2 3 1 4 5 2 3 1 4 4 5 2 3 1 4 1 4 5 5 2 3 5 2 3 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

Let Let G G be be a split graph a split graph with with even even maximum maximum degree degree. .

∆(G)=10

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

Construct Construct a a commutative commutative latin latin square square of

• f order
• rder

∆ ∆(G) (G)-

• 1

1, where m , where mi,j

i,j =

= i+j i+j ( (mod mod ∆

∆(G)

(G)-

• 1

1) )

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1 8 6 5 4 3 2 1 8 7 5 4 3 2 1 8 7 6 4 3 2 1 8 7 6 5 3 2 1 8 7 6 5 4 2 1 8 7 6 5 4 3 1 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

Let Let’ ’s use this s use this commutative commutative latin latin square square to to construct construct: :

• a

a matrix matrix A ( A (that that we we use to color G[Q]) use to color G[Q]) and and

• a

a monotonic monotonic color color diagram diagram D ( D (that that we we use to color a use to color a bipartite bipartite graph graph B=[Q,S]). B=[Q,S]).

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1 8 6 5 4 3 2 1 8 7 5 4 3 2 1 8 7 6 4 3 2 1 8 7 6 5 3 2 1 8 7 6 5 4 2 1 8 7 6 5 4 3 1 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

|Q|=7

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1 8 6 5 4 3 2 1 8 7 5 4 3 2 1 8 7 6 4 3 2 1 8 7 6 5 3 2 1 8 7 6 5 4 2 1 8 7 6 5 4 3 1 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 5 4 3 2 1 8 7 6 7 6 5 4 3 2 1 8 6 5 4 3 2 1 8 7 4 3 2 1 8 7 6 5 3 2 1 8 7 6 5 4 2 1 8 7 6 5 4 3 1 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 8 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

d(S)=5

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 7 6 5 4 3 2 1 8 4 3 2 1 8 7 6 6 5 4 3 2 1 8 5 4 3 2 1 8 7 3 2 1 8 7 6 5 2 1 8 7 6 5 4 1 8 7 6 5 4 3 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 7 6 4 3 3 2 1 8 5 3 3 2 1 8 7 6 6 5 4 3 2 1 8 5 4 3 2 1 8 7 4 1 2 1 8 7 6 5 2 1 8 7 6 5 4 1 8 7 6 5 4 3 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 9 9 9 9 9 9 9 4 3 3 2 1 8 5 3 3 2 1 8 7 6 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 6 5 2 1 8 7 6 5 4 1 8 7 6 5 4 3 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 9 9 9 9 9 9 9 4 3 3 2 1 8 5 3 3 2 1 8 7 6 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 6 5 2 1 8 7 6 5 4 1 8 7 6 5 4 3 8 7 6 5 4 3 2 8 7 6 5 4 3 2 1 7 6 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

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Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 6 6 9 9 9 6 6 4 3 3 2 1 8 5 3 3 2 1 8 7 9 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 9 5 2 1 8 7 6 5 4 1 8 7 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 6 6 9 9 9 6 6 4 3 3 2 1 8 5 3 3 2 1 8 7 9 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 9 5 2 1 8 7 6 5 4 1 8 7 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 6 6 9 9 9 6 6 4 3 3 2 1 8 5 3 3 2 1 8 7 9 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 9 5 2 1 8 7 6 5 4 1 8 7 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 6 6 7 7 9 6 6 4 3 3 2 1 8 5 3 3 2 1 8 7 9 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 9 5 2 1 8 9 6 5 4 1 8 9 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 6 5 6 6 7 7 9 6 6 4 3 2 2 1 8 5 3 3 2 1 8 7 9 7 6 4 1 2 1 8 7 9 5 3 8 1 8 9 6 5 4 1 8 9 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

9 5 4 3 2 1 3 4 5 9 7 9 8 5 6 7 8 1 2

1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 6 5 6 6 7 7 9 6 6 4 3 2 2 1 8 5 3 3 2 1 8 7 9 7 6 4 1 2 1 8 7 9 5 3 8 1 8 9 6 5 4 1 8 9 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

9 5 4 3 2 1 3 4 5 9 7 9 8 5 6 7 8 1 2

1 2 3 4 5 6 1 2 4 5 6 6 7 7 9 6 6 4 3 2 2 1 8 5 3 4 1 3 8 1 8 7

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 1 2 4 5 6 6 7 7 9 6 6 4 3 2 2 1 8 5 3 4 1 3 8 1 8 7

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

9 5 4 3 2 1 3 4 5 9 7 9 8 5 6 7 8 1 2

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 6 6 7 7 9 6 6 4 3 3 2 1 8 5 3 3 2 1 8 7 9 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 9 5 2 1 8 9 6 5 4 1 8 9 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1 Color of the element m(|Q|+1)/2,(|Q|+1)/2

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 6 6 7 7 9 6 6 4 3 2 1 1 8 5 3 3 2 1 8 7 9 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 9 5 3 8 1 8 9 6 5 4 2 6 8 9 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8 6 5 6 6 7 7 9 6 6 4 3 2 1 1 8 5 3 3 2 1 8 7 9 7 5 4 3 2 1 8 6 4 3 2 1 8 7 4 1 2 1 8 7 9 5 3 8 1 8 9 6 5 4 2 6 8 9 6 5 4 3 8 7 6 5 4 3 2 8 7 9 5 4 3 2 1 7 9 5 4 3 2 1

Necessary condition: there is a vertex in S with degree at least |Q|/2

Using Latin Squares to Color Using Latin Squares to Color Split Graphs Split Graphs

Necessary condition: (d(Q))2 ≥ 2|Q|+1

Number of times that the color (∆(G)-1) can appears in the monotonic color diagram is at most d(Q)-1, so: |Q|-(∆-|Q|+2(∆-|Q|-2)+(∆-|Q|-4)+ ... +(∆-|Q|-(∆-|Q|-2)) = |Q|-(d(Q)-1+2(d(Q)-3+ ... +d(Q)-(d(Q)-2))) = |Q| - (d(Q) -1 + ( (d(Q)-1) (d(Q)-3) )/2 = |Q| - ((d(Q)-1)2)/2 |Q| - ((d(Q)-1)2)/2 ≤ d(Q)-1 2|Q| - (d(Q)-1)2 ≤ 2d(Q)-2 2|Q| ≤ 2d(Q)-2 + (d(Q)-1)2 |Q| ≤ (d(Q))2-1 (d(Q))2 ≥ 2|Q|+1

The Classification Problem for Split The Classification Problem for Split Graphs Graphs with with even even maximum maximum degree degree

Let Let G G be be a split graph a split graph with with even even maximum maximum degree

• degree. If G

. If G has has a a vertex vertex in S in S with with degree degree at at least least | |Q|/2 Q|/2 and and d(Q) d(Q)2

2 ≥

≥ 2| 2|Q|+1 Q|+1, , then then G G is is Class Class 1. 1.

Thank you