On the Linear K-Arboricity of Regular Graphs

More on the Linear k-arb oricity of Regular

Graphs

R. E. L. Aldred

Department of Mathematics and Statistics

University of Otago

P.O. Box 56, Dunedin

New Zealand

Nicholas C. Wormald

Department of Mathematics

University of Melb ourne

Parkville, VIC 3052

Australia

Abstract

Bermond et al. [2] conjectured that the edge set of a cubic graph G

can b e partitioned into two linear k -forests, that is to saytwo forests

whose connected comp onents are paths of length at most k , for all

k 5. That the statement is valid for all k 18 was shown in [5 ] by

Jackson and Wormald. Here we improve this b ound to

7 if G=3;

9 otherwise.

The result is also extended to d-regular graphs for d > 3, at the

exp ense of increasing the numb er of forests to d 1.

All graphs considered will b e nite. We shall refer to graphs which may

contain lo ops or multiple edges as multigraphs and reserve the term graph

for those which do not. A linear forest is a forest each of whose comp onents 1

are paths. The linear arboricity of a graph G, de ned by Harary [4], is the

minimum numb er of linear forests required to partition E G and is denoted

by laG. It was shown by Akiyama, Exo o and Harary [1] that laG = 2

when G is cubic. A linear k -forest is a forest consisting of paths of length

at most k . The linear k -arboricity of G, intro duced by Bermond et al. [2],

is the minimum number of linear k -forests required to partition E G, and

is denoted by la G. When such a partition of E G has b een imp osed, we

say that G has b een factored into linear k -forests. We refer to each linear

forest in such a partition of E G as a factor of G and the partition itself is

called a factorization.

It is conjectured in [2] that if G is cubic then la G 2. A partial

result is obtained by Delamarre et al. [3] who show that la G 2 when

k jV Gj 4. Jackson and Wormald [5] improved this result when

jV Gj 36, showing that, for k 18, an integer, la G= 2. Here we shall

improveon this further to show the following.

Theorem. Let G b e a cubic graph and let k b e an integer. Then la G= 2

for all

7 if G= 3;

9 otherwise.

Before we prove the theorem, we shall present some preliminary results

which indicate some restrictions we can demand of factorizations of cubic

graphs. These will b e most useful in the pro of of the theorem.

For our rst result weintro duce the following terminology. An odd linear

forest is a linear forest in which each comp onent is a path of odd length.

Also, we use G to denote the chromatic index of G i.e. the minimum

number of colours required to colour the edges of G so that no two edges of

the same colour are incident with the same vertex.

Lemma. Let G be a cubic graph. Then G can be factored into two odd

linear forests if and only if G=3.

Pro of. Supp ose rst that G is a cubic graph with a factorization, F ;F

1 2

into odd linear forests F and F . Colour the edges of the paths in F al-

1 2 1

ternately red and blue so that each path in F has its rst and last edges

coloured red. Similarly, colour the edges of the paths in F alternately green

and blue so that each path in F has its rst and last edges coloured green.

This yields a prop er 3-edge-colouring of G giving G= 3. 2

Conversely, let us supp ose that G = 3 and that we have a prop er

3-edge-colouring imp osed on G using the colours blue, green and red. Let

0 0

F and F be factors of G induced by the blue and green edges and by the

1 2

0 0

red edges resp ectively. Thus F consists of disjoint even cycles, while F is

1 2

a set of disjoint edges covering the vertices of G. Form new factors F and

F where F is the subgraph of G induced by the edges in F together with

2 2

at most one edge from each cycle in F chosen so that F is acyclic and such

that the paths in F have maximum p ossible total length. The factor F

2 1

consists of F with the edges in F removed.

Claim: F contains an edge from each cycle in F . To see this, assume

to the contrary that there is a cycle C = v v :::v v in F . Then v and v

1 2 k 1 1 1 2

are b oth ends of the one path in F , while v and v are b oth ends of the one

2 2 3

path in F by the maximality of total length of paths in F . Since G is a

2 2

graph, v 6= v and no path has three ends so the claim follows.

1 3

Thus each path in F has o dd length and each path in F has o dd length

1 2

these paths must b egin and end with red edges and haveevery second edge

red throughout their lengths.

Corollary. Let G be an r -regular graph. If G = r 1, then G can be

factored into r 1 odd linear forests.

Pro of. Since G=r , the edges of G can be partitioned into r 1-factors.

cho ose r 3 of these to form factors in our factorization, the remaining 3

1-factors inducing a cubic spanning subgraph of G which has chromatic index

3. By the lemma, this subgraph can be factored into two o dd linear forests,

completing the desired factorization and the pro of of the corollary.

Prop osition. Let G be a cubic graph. Then there is a factorization of G

into two linear forests, F and F such that no comp onent in F isomorphic

1 2 i

to P has its end vertices adjacent in G via an internal edge of a path in

F , where fi; j g = f1; 2g. Such a factorization is to have the spread-P

j 3

prop erty.

Pro of. The prop osition is certainly true for G = K . So assume that G is a

cubic graph of smallest order which do es not admit a factorization with the

spread-P prop erty. Clearly, G cannot b e triangle-free. But if we have a tri-

angle T , with each edge b elonging to precisely one triangle, then contracting

T to a vertex yields a cubic graph, G , of smaller order than than G. By the

minimalityof G, G admits a factorization with the spread-P prop erty. This

3 3

factorization easily lifts to a factorization of G with the spread-P prop erty.

Consequently, we may assume that all triangles in G are paired. By [1]

we can let G b e factored into two linear forests. Assume this is done so that

there are as few unspread-P 's as p ossible. By unspread-P we mean a path

3 3

of length two in one factor whose endvertices are adjacentinG via an internal

edge in the other factor. As G was chosen not to admit a factorization with

the spread-P prop erty, we must have an unspread-P in one of the paired

3 3

triangles. Figure 1 indicates how this factorization may b e altered to reduce

the number of unspread-P 's. This contradiction establishes the result.

. .

@ @

. .

@ @

. - . .

@ @

Figure 1.

Pro of of Theorem. De ne F = F G to be the set of F ;F such that

1 2

F and F form a factorization of G with the spread-P prop erty. Then F

1 2 3

is nonempty by the prop osition. For F ;F 2 F , let l F ;F denote the

1 2 1 2

length of the longest path in F or F . Let t be the minimum number such

1 2

that there exists F ;F 2F with l F ;F = t.

1 2 1 2

Cho ose F ;F 2 F with l F ;F = t and such that, sub ject to this

1 2 1 2

condition, the total number mF ;F , of paths of length t in F and F is

1 2 1 2

as small as p ossible. By symmetry, we may assume that F has a path P of

length t.

Assume t 10. We will use the structure of this factorization to pro duce

distinct edges, e of G for each integer i 0, thereby contradicting the

niteness of G. To assist in this pro cess we de ne an i-improvement.

Given path sets S ;:::;S , de ne R = R S ;:::;S to be the set of all

0 i i i 0 i

edges of G which are either contained in or incident with any of the paths in

S ;:::;S .

0 i

0 0

Given F ;F 2F and path sets S ;:::;S , we say that F ;F 2F is

1 2 0 i

1 2

an i-improvement of F ;F if there exists an edge e of a path in S such

1 2 i

that

i they are in F , i all edges in R n e are in F

1 i

1 4

ii e is in F i it is not in F , and

iii each path in F of length at least t is also a path in F , j = 1 and 2.

Wenow describ e the algorithm for pro ducing the edges e referred to as

growth edges. The algorithm pro ceeds through steps of cho osing a growth

edge and \growing" one or two paths from its endvertices. The rst path

\grown" is P . Certain vertices b ecome \p oison" and later \kill" the \live"

edges, which are the p otential growth edges.

Initially we set:

S := fP g denoting the set of one or two paths just \grown";

T := a maximum set of indep endent internal edges in P excluding those

third from either end \live" edges, from which it is p ossible to cho ose

growth edges;

W := fend vertices of P g \p oison" vertices.

Note that when t 5, jT j = d e. So for t 7we have have jT j > jW j.

0 0 0

For i>0 carry out the following three steps.

1. Cho ose e 2 T .

i1 i1

2. Rep eat a and b consecutively until b requires no action b ecause its

condition fails:

a De ne S to b e the set of paths which terminate at the endvertices

of e .

0 0

b If there is an i-improvementF ;F ofF ;F then reset F ;F

1 2 1 2

1 2

0 0

. ;F to b e equal to F

2 1

3. Determine the sets T and W by the following metho d. Set:

i i

V := f endvertices of paths in S gnfendvertices of e g;

i i i1 5

N := a maximum set of indep endent internal edges of paths in S

i i

other than those third along from a vertex in V ;

T := T nfe g [ N adding edges in N to the set of live edges,

i1 i1 i i

and deleting e ;

W := W [ V V contains new p oison vertices;

i1 i i

0 0 0

T := T nfe 2 T incident with a vertex in W g these edges are

i i i

p oisoned;

0 0 0

W := W nfw 2 W incident with and edge in T g after a vertex

i i i

p oisons an edge, it is no longer p oison.

Step 2 must eventually terminate b ecause the rede nition of F ;F in

1 2

2b leads to a strictly smaller total length of paths in the set S when 2a

is next p erformed.

Denote the value of F ;F at the end of the ith iteration of the two

1 2

steps ab ove by F ;F . Then for any j >i the set S retains its prop erty

i;1 i;2 i

of b eing a set of maximal paths in F or F , b ecause the rede nition of

j;1 j;2

F ;F do es not disturb any edges in R . Thus, F ;F cannot have

1 2 j 1 j;1 j;2

any i-improvement for i

Note also that prosp ective growth edges in N are p oisoned by vertices

precisely when such an edge shares an endvertex with a path in some in W

S ;j i 1. In this way, the paths in S cannot b e included in S ;j i 1.

j i j

whenever Thus an edge e 2 T chosen in Step 1 cannot b e contained in T

h h h

h >h.

Consequently, having chosen e 2 T for Step 1, we have one of two

i1 i1

p ossiblities at the completion of Step 2.

i There is a single path, P ,inS joining the endvertices of e . Since we

i i1

are working with factorizations with the spread-P prop erty, P must

have length at least 3. As we work through Step 3 in this case, V = ;

and we have at least one edge in N . Each edge of T \killed" by live

results in one vertex from W not b eing retained in W vertices in W

i1 i

and, in this case, no new vertices will b e added to W when forming

W since V = ;. Thus jT j increases by at least as much as jW j.

i i i i

~ ^

ii There are two paths P; P 2 S terminating at the endvertices of e .

i i1

~ ^

Supp ose that l P +l P t 2. Then e cannot b e in S = P if it

i1 0 6

were, then swapping e between the appropriate factors would pro-

duce a factorization in F in which the total numb er of paths of length

at least t is less than in our originally chosen factorization; a contra-

diction. Thus e is in some path in S ; 1 j

i1 j

e between the appropriate factors will yield a j -improvement for

some j < i. The internal edges third from a "live" end of a path in

S ; 0 j i 1 have b een excluded from T to ensure that swap-

j i1

ping e between factors cannot result in an unspread-P , pro ducing

i1 3

~ ^

a factorization not in F . . Thus we have l P +l P t 1.

Working through Step 3 in this case, it can b e checked that N contains

at least three edges, as t 10. As b efore, each edge of T \p oisoned"

0 0

by a vertex in W results in one vertex from W not b eing retained in

i i

W . There are at most two new vertices added to W to form W and

i i1

at least three new edges added to T to obtain T . Once again, jT j

i1 i

increases by at least as much as jW j.

In b oth cases we see by induction that jT j the number of live edges

exceeds jW j the number of p oison vertices. Thus we may continue inde -

nitely, removing a di erent edge from T at each iteration. This contradicts

the niteness of G.

Next assume that G = 3. By the Lemma, G can be factored into

two o dd linear forests. Working through the pro of ab ove restricting F to the

set of factorizations of G into two odd linear forests, we make the following

minor mo di cations.

T := every second edge in P ;

N := every second edge in paths in S .

i i

~ ^

Now, everything follows as b efore since in case ii ab ove, the paths P; P are

restricted to be of odd length. Consequently, the niteness of G is contra-

dicted if t 9, as all paths are of odd length and thus t, the length of a

longest path in either factor, must also b e o dd. The result follows.

Finally we note that our Theorem can easily be extended to graphs of

maximum degree three since any such graph can b e emb edded as a subgraph

of a cubic graph. Using this we deduce the following: 7

Corollary. Let G be a 4-regular graph and k 9 be an integer. Then

la G=3.

Pro of. Using [7] wemaycho ose a 2-factor F in G. Clearly F has a spanning

k -linear forest D . Since H = G E D has maximum degree three, it follows

from the ab ove-mentioned extension of our Theorem that la H =2. Thus

la G=3.

In [6], Lindquester and Wormald considered the following variation of

linear arb oricity for r -regular graphs. An r -regular graph G is said to be

l; k -linear arbori c if it can be factored into l linear k -forests. In this

setting, our results enable us to prove the following.

Corollary. Let G be an r -regular graph, 3, and let k b e an integer. Then

G is r 1;k-linear arb ori c for all

7 if G= r ;

9 otherwise.

Pro of. If G = r , then we may factor G into r 1-factors. Cho ose r 3

of these 1-factors and delete them leaving a cubic spanning subgraph of G

which is 3-edge-colourable. By our theorem, this subgraph is 2; 7-linear

arb ori c and thus G is r 1; 7-linear arb ori c as required.

When G 6= r , we pro ceed by induction on r . Our theorem and the

ab ove corollary show that this is true for r =3; 4. Assume that the statement

is true for all r r and let G be an r + 1-regular graph. If r +1 is even,

0 0 0

then G has a 2-factor, F , say. The graph G F is regular of degree r 1

and so, by induction, may b e factored into r 2 linear 9-forests. The desired

factorization is completed by decomp osing F into two linear 2-forests in the

obvious way.

So assume that r +1 is o dd. Let M be a maximum matching in G.

Then G M has vertices of degrees r and r + 1, and the vertices of degree

0 0

r + 1 form an indep endent set, B ,say. The subgraph of G M induced by

B [ N B with edges between vertices in N B deleted is bipartite and, by

Hall's theorem and considerations of degrees, contains a matching, M , which

0 0

covers the vertices in B . Thus G = G M [ M is biregular with degrees

0 0 00

r and r 1. If H is an isomorphic copyofG and G is the graph obtained

0 0

from G and H by adding a matching b etween the vertices of degree r 1

0 0 00

in G and the vertices of degree r 1 in H , then G is r -regular and, by

0 0

our inductivehyp othesis, contains a factorization into r 1 linear 9-forests.

0 8

Restricting these factors to the vertices in G together with M [ M gives the

desired factorization of G into r linear 9-forests. M [ M is a linear forest

with comp onents of size 1 and 2.

References

[1] J. Akiyama, G. Exo o and F. Harary, Covering and packing in graphs

III, cyclic and acyclic invariants, Math. Slovaca, 30 1980 405-417.

[2] J.C. Bermond, J.L. Fouquet, M. Habib and B. Pero che, On linear k-

arb oricity, Discrete Math., 52 1984 123-132.

[3] D. Delamarre, J.L. Fouquet, H. Thuillier and B. Virot, Linear k-

arb oricity of cubic graphs, preprint. Magyar Tud.

[4] F. Harary,Covering and packing in graphs I, Ann. New York Acad. Sci.,

175 1970 198-205.

[5] Bill Jackson and Nicholas C. Wormald, On the linear arb oricity of cubic

graphs, to appear in Discrete Math..

[6] T. Lindquester and Nicholas C. Wormald, Linear arb oricityof r -regular

graphs, submitted.

[7] J. Petersen, Die Theorie der regularen Graphen, Acta Math., 15 1891,

193-220. 9