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On the edgecolouring of

graphs

Celina M H de Figueiredo Joao Meidanis

y

Celia Picinin de Mel lo

Relatorio Tecnico IC Junho de

On the edgecolouring of split graphs

y y

Celina M H de Figueiredo Joao Meidanis Celia Picinin de Mello

Abstract We consider the following question can split graphs with o dd maximum

b e edgecoloured with maximum degree colours We show that any o dd maxi

mum degree can b e transformed into a sp ecial split graph For this sp ecial

split graph we were able to solve the question in case the graph has a quasiuniversal

Sumario Consideramos o seguinte problema grafos split com grau maximo mpar

admitem uma coloracao de arestas com grau maximo cores Mostramos como qualquer

grafo split com grau maximo mpar p o de ser transformado em um grafo split esp ecial

Para este grafo split esp ecial resolvemos o problema prop osto no caso do grafo admitir

um vertice quaseuniversal

Intro duction

An edgecolouring of a graph is an assignment of colours to its edges such that no adjacent

edges have the same colour The chromatic index of a graph is the minimum numb er of

colours required to pro duce an edgecolouring for that graph

An easy lower b ound for the chromatic index is the maximum vertex degree A

celebrated theorem by Vizing states that these two quantities dier by at most one

Graphs whose chromatic index equals the maximum degree are said to b e Class graphs

whose chromatic index exceeds the maximum degree by one are said to b e Class

Very little is known ab out the complexity of computing the chromatic index in general

It is known that complete graphs and chordless cycles are Class if and only if the numb er

of vertices is even Bipartite graphs are all Class Planthold proved that graphs

with a or with a quasiuniversal vertex can b e classied in p olynomial

time Homan and Ro dger derived a similar result for complete multipartite graphs

The fourcolour problem is equivalent to saying that all connected regular planar graphs

are Class In previous pap ers we settled the question for doubly chordal graphs with o dd

maximum degree and for indierence graphs with at most three maximal cliques

Doubly chordal graphs include interval graphs and strongly chordal graphs These results

summarize all that is known ab out edgecolouring sp ecic classes of graphs

In this note we consider the following question can split graphs with o dd maximum

degree b e edgecoloured with colours Edgecolouring of split graphs was rst considered

Instituto de Matematica Universidade Federal do Rio de Janeiro Caixa Postal Rio

de Janeiro RJ Brasil celinacosufrjbr Partially supp orted by CNPq

y

Instituto de Computacao Universidade Estadual de Campinas Caixa Postal Campinas

SP Brasil fmeidanisceliagdccunicampbr Partially supp orted by FAPESP and CNPq

On the edgecolouring of split graphs

in In that work we discussed lo cal sucient conditions for a graph to b e Class We

considered sub classes of chordal graphs such as split graphs and interval graphs

We b egin this note characterizing two sub classes of split graphs which we call easy and

difficult The easy ones can b e easily coloured by a pullback technique We essentially

b orrow the colour from a The difficult ones as the name indicates

are hard to colour but this class has the imp ortant prop erty that if all its memb ers are

Class then al l o dd maximum degree split graphs are Class

Next we show that every difficult graph can b e transformed into an easy one by

rep eated applications of a sub division op eration on vertices The idea is to use this trans

formation to transfer the colouring of the easy graph to the difficult one The transfor

mation increases the numb er of vertices but keeps the numb er of edges constant We were

able to colour some difficult graphs with this technique and we show these results in the

sequel Sp ecically we decided the cases where one sub division op eration is enough

Denitions and notations

General terms

In this pap er G denotes a simple undirected nite connected graph V G and E G are

the vertex and edge sets of G A stable set is a set of vertices pairwise nonadjacent in G

A is a set of vertices pairwise adjacent in G A maximal clique of G is a clique not

prop erly contained in any other clique A maximum clique of G is a clique of maximum

size ie a clique with the largest p ossible numb er of vertices

A subgraph of G is a graph H with V H V G and E H E G For X V G

we denote by GX the subgraph induced by X that is V GX X and E GX consists

of those edges of E G having b oth ends in X

For each vertex v of a graph G Adj v denotes the set of vertices that are adjacent to v

In addition N v denotes the neighbourhood of v that is N v Adj v fv g The degree

of a vertex v is deg v jAdjv j The maximum degree of a graph G is then G

max deg v A vertex u is universal if deg u jV Gj A vertex u is quasi

v V (G)

k

universal if deg u jV Gj Given a graph G and k we denote by G the graph

k k

having V G V G and satisfying xy E G if and only if x and y are distinct and

their distance in G is at most k The diameter of G is diamG max distv w

v w V (G)

K denotes the complete graph on n vertices

n

A split graph is a graph whose vertex set admits a partition into a stable set and a

clique

Colouring

An assignment of colours to the vertices of G is a function V G S The elements

of the set S are called colours A conict in an assignment of colours is the existence of

two adjacent vertices with the same colour A vertexcolouring of a graph is an assignment

of colours such that there are no conicts The chromatic number of a graph G is the

minimum numb er of colours used among all vertexcolourings of G and is denoted by G

On the edgecolouring of split graphs

An assignment of colours to the edges of G is a function E G S Again the

elements of the set S are called colours A conict in an assignment of colours is the

existence of two edges with the same colour incident to a common vertex A vertex u is

said to b e satised when uv uw implies v w for all neighb ours v w of u An

edgecolouring of a graph is an assignment of colours such that every vertex is satised or

equivalently such that there are no conicts The chromatic index of a graph G is the

minimum numb er of colours used among all edgecolourings of G and is denoted by G

A graph G is said to b e Class if G G and Class if G G

Vizings theorem states that there are no other p ossibilities all graphs are either Class

or Class A constructive pro of of Vizings theorem app eared in

easy and difficult

Recall that split graph is a graph G for which V G can b e partitioned into two sets A and

B such that A is a clique and B is a stable set A splitting clique A of G is a maximal

clique such that V G n A is a stable set of G Note that every splitting clique is a clique

of maximum size in G

We concentrate on split graphs of o dd maximum degree as mentioned in the intro

duction We now dene a sub class of o dd maximum degree split graphs called easy which

contains all graphs we consider easy to colour with colours An o dd maximum degree

2

split graph G is said to b e easy if G G

Theorem easy Class

The pro of is based on the concept of a pullback function A pul lback from graph G to

graph H is a function f V G V H such that

f is a homomorphism ie f takes adjacent vertices of G into adjacent vertices of H

f is injective when restricted to each neighb ourho o d

The following results relate edgecolouring to pullbacks They are proved in

Theorem If f is a pul lback from G to H and H is k edgecolourable then G is k edge

colourable

Theorem Composition of pul lbacks is a pul lback

2

Any graph satisfying G G admits a pullback into the complete graph

K In particular easy graphs can b e coloured with colours b ecause they admit a

+1

pullback into the graph K a complete graph with an even numb er of vertices

+1

We now dene the sub class of difficult graphs An o dd maximum degree split graph

G is said to b e difficult when

all vertices in a splitting clique A of G have degree G

diamG

On the edgecolouring of split graphs

The imp ortance of this class is justied by Theorem b elow Together with Theorem

they imply that if we prove that difficult Class then all o dd maximum degree split

graphs are Class

Theorem If G is any odd maximum degree split graph then there is a difficult graph

H with G H and a pul lback function f V G V H

Pro of Supp ose G is a split graph with diamG Let V G b e partitioned into

a splitting clique A and a stable set B If there is a vertex v in A with deg v

then we add an edge b etween v and any vertex u not adjacent to v This addition do es

not change the maximum degree b ecause deg u If u now sees every vertex of A

then A A fug is now a splitting clique with corresp onding stable set B B n fug By

rep eating this op eration as many times as necessary we eventually obtain a graph satisfying

the rst condition in the denition of difficult

If at this p oint the diameter is greater than we take any two vertices x and y with

distx y and join them into a single vertex z If vertex z sees every vertex of A

then A A fz g is now a splitting clique with corresp onding stable set B B n fz g If

necessary we add edges incident to z until we obtain a graph satisfying the rst condition

in the denition of difficult

Furthermore there is a pullback function from the old graph into the new one that maps

x and y into z and maps the other vertices into themselves By rep eating this op eration as

many times as necessary we eventually obtain a graph with diameter at most The nal

pullback is just the comp osition of all the previous pullbacks

Transforming difficult into easy

Given a graph G and an integer k we dene r emainG k as the least numb er of vertices

of G whose removal leaves a graph that admits a vertexcolouring with k colours Note that

2

if G is an easy graph then r emainG On the other hand given a difficult

2

graph G such that r r emainG the removal of any stable set of size r from

2

G leaves a graph H with r emainH

Given a graph G a vertex subdivision op eration transforms G into a graph sG and

is dened as follows Let v V G and let Adj v b e partitioned into sets V V In

1 t

sG we have one new vertex v for each set V The vertex and edge sets are dened as

i i

V sG V G n fv g fv v g and E sG E G n fv w w Adj v g fv u

1 t i i

u V g We say that vertex v is subdivided into vertices v v

i i 1 t

The main result of this section shows how to transform a difficult graph into an easy

graph

Theorem For every difficult graph there is a series of subdivision operations that

transforms it into an easy graph

On the edgecolouring of split graphs

2

Pro of Let G b e a difficult graph with r emainG We shall prove that there

2

is a sub division op eration that transforms G into sG such that r emainsG

2

r emainG

2

Let r r emainG By denition there exists a set R of r vertices of G

2

whose removal leaves a graph G such that G admits a vertexcolouring C with

colours Let v R Dene the following subsets of Adj v

P r eV fn Adj v C n i n Adj w C w ig

i

We note that Adj v P r eV Indeed if x Adj v but x P r eV then N x C

i i i

for i and Adj x R This implies deg x a contradiction

Some sets P r eV are p ossibly empty and the sets P r eV are not necessarily disjoint Now

i i

there exists a collection of sets V V that partitions Adj v and such that V P r eV

1 t j i

for i Moreover there are at least two distinct indices i and j with V and V

i j

not empty If only one V is not empty then it is p ossible to colour v Then this partition

i

denes a sub division of vertex v which in turn denes the required graph sG

When the graph is almost easy

Let G b e a difficult graph with the usual partition of V G into sets A and B Let jAj a

2

and jB j b In this section we solve the case r emainG We show how to

use a sub division op eration together with a pullback to get the required edgecolouring of

G with colours

Let v b e any vertex of B Lab el the vertices of B n v with lab els s We note that

we have the following relations s b a b a b

As in the pro of of Theorem sub divide vertex v into vertices v v The set Adj v

1 s

is partitioned into V V accordingly We note that graphs G and sG have the same

1 s

numb er of edges In order to obtain the required edgecolouring of G we shall rst obtain

an edgecolouring of sG with colours

Now we pro ceed to exhibit a pullback from sG to K

+1

Vertices of B n v continue to have lab els s Now lab el v with lab el i i s

i

and lab el vertices in V V with lab els s s s deg v following the ordering

1 s

of the subscripts of the sets V

i

Note that s b b ecause v B On the other hand deg v a b ecause no

vertex of B sees all vertices of A Hence s deg v a b Finally to nish the

pullback lab el the vertices of A n Adj v with lab els not used so far

To see that this edgecolouring of sG is in fact an edgecolouring of G consider the

edge joining vertex x V to vertex v Let us compute the size of the interval of the sums

i i

l abel x i where x V and i s The minimum p ossible is s s

i

The maximum p ossible is s s deg v s deg v Hence this interval has size

s deg v s s deg v Now s deg v implies that the size of this

interval is at most In particular we get that these values b elong to distinct classes

mo dulo

Therefore we have the following theorem

On the edgecolouring of split graphs

Theorem Let G be a split graph with odd maximum degree Suppose that G satises

2

r emainG Then G is Class

Conclusions

We have rep orted our progress on edgecolouring o dd maximum degree graphs Since it is

NPcomplete to decide whether an o dd maximum degree graph is Class we have b een

solving the problem for sp ecial classes doubly chordal graphs and indierence graphs

We conjectured in that every o dd maximum degree split graph is Class

In this note we have given p ositive evidence for this conjecture by proving it for sp ecial

cases Our metho d denes two functions sub division which maps a split graph into a

particular Class split graph and pullback a sp ecial kind of homomorphism

Acknowledgements This work was done while the authors were visiting the Department

of Combinatorics and Optimization at the University of Waterlo o Canada The rst author

holds a p ostdo ctoral grant from CNPq Pro c The second and third authors

acknowledge supp ort from CNPqProTeMCCI I Pro ject ProComb Pro c

and FAPESP

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