Discrete Applied Mathematics 27 (1990) 287-292

North-Holland

287

COMMUNICATION

FRACTIONAL TOTAL COLOURING

Jennifer RYAN

Department of Mathematics, University of Colorado, Denver, CO 80204, USA

Received 3 May 1989

Communicated by P. Hansen

Behzad conjectured in 1965 that a simple graph G can always be totally coloured using two

more colours than the maximum degree in G, d(G). If the conjecture is true, then the total

colouring number of a graph is always equal to either d(G)+ 1, or d(G) +2. A weakening of

Behzad’s conjecture is proved here by presenting an integer programming formulation of the pro-

blem and showing that its linear programming relaxation has its value bounded between d(G) + 1

and d(G) + 2.

1. Introduction

Definition. A total colouring of a graph is a colouring of the vertices and edges such

that no two adjacent or incident objects receive the same colour. The total colouring problem is to totally colour G using as few colours as possible.

Behzad [l] conjectured in 1965 that a graph G can always be totally coloured

using d(G) + 2 colours, where d(G) is the maximum degree of a vertex in G. It is

clear that the minimum number of colours needed is d(G) + 1, since the vertex

having maximum degree and all its incident edges must receive distinct colours. The

conjecture has been verified for certain special cases, [2,3,5,6,8], but a general

proof has been elusive.

The total colouring problem bears some similarity to the edge colouring problem.

Vizing’s theorem, (see e.g. [4]), states that a graph can always be edge coloured with

d(G) + 1 colours. Since it is clear that at least d(G) colours will always be needed

to edge colour a graph, we have that the edge colouring number of a graph is always

either d(G) or d(G) + 1. If the total colouring conjecture is true, then the total

colouring number of a graph is always either d(G) + 1, or d(G) + 2. Further, there

is an integer programming formulation of the edge colouring problem whose linear

programming relaxation has its value bounded between d(G) and d(G) + 1 (see e.g.

[7]). An integer programming version of the total colouring problem is presented

here whose linear programming relaxation has its value bounded between d(G) + 1

and d(G)+2.

0166-218X/90/$03.50 0 1990 - Elsevier Science Publishers B.V. (North-Holland)

288 J. Ryan

2. The fractional total colouring problem

Let G be a graph with vertices I/ and edges E. For a vertex i E V, let 6(i) denote

the set of edges having i as an endpoint and let deg(i) = IS(i)1 be the degree of i. An

edge e E E may be denoted (i, j) where i and j are its endpoints.

2.1. The integer programming formulation

We begin by defining the variables. Let yk, i = 1, . . . , m, be defined

l

1,

yk= 0,

if colour i is used,

otherwise.

The integer m must be chosen large enough so that the problem is known to be feasi-

ble. (For instance, m = 1 I// + lE 1 would work trivially.) For each eE E and each

k= 1, . ..) m define

1

1,

zek= 0,

if edge e is coloured with colour k, otherwise.

For each i E V and each k = 1,. . . , m define

L

1, if vertex i is coloured with colour k,

x’k= 0, otherwise.

The first constraint in the formulation is that given iE V, the vertex i and all of

its incident edges must receive distinct colours. Thus for all ie V and for all

k= 1, . . . . m we have the constraint

c z,k + X;k 5 I. (1) es&i)

Also, no two adjacent vertices can receive the same colour so, for all e = (i, j) E E, and for all k=l,...,m we must have

x,+xjk< 1. (2)

In order to force yk = 1 if colour k is used we add the constraints

and

(3)

(4)

Finally we insist that each vertex and each edge receive at least one colour. For all

iE V,

kE, Xik 2 l, (9

Fractional total colouring 289

and for all eeE,

ki!, &k z l.

Thus the total colouring problem is the integer programming problem

(6)

minimize kE, yk)

sub_iect to Cl), (21, (3), (4), (5) and (6), y,E{O,l}, k=l,..., m,

X;kE{O, 1}, Vie v, k=l,..., m,

Z&E{O,l}, VeeE, k=l,..., m.

2.2. Bounds on the value of the relaxation

The linear programming relaxation of (7) is

minimize ,t, yk )

subject to Cl), Q), (3), (4), (5) and (6), yk?O, k=l,..., m, x,krO, vie V, k=l,..., m, z&rO, VeeE, k=l,..., m.

(7)

(8)

Note that it is not necessary to restrict the variables to be less than or equal to 1;

this will be enforced by the constraints (1) and (2) and by the minimization.

Proposition 1. The value of (8) is at least d(G) + 1.

Proof. Let I be the vertex having maximum degree. Then for each k= 1, . . . , m,

Yk>c eE6C,j i&k +x/k, by (3) with i= 1. Then, using also (5) and (6),

rdeg(l)+ 1 =A(G)+ 1. 0

To get an upper bound on the value of (8), its linear programming dual will be

considered. Let W;k 5 0 be the dual variable corresponding to the i, k constraint of

(l), for all ie V and k= 1, . . . . m. For all eEE and k= 1, . . . . m, let q&IO be the

dual variable corresponding to the e, k constraint of (2). Define Uik and u& for the

(3) and (4) constraints similarly. Finally let ri 2 0 be the dual variable for constraint

i of (5), and s,>O be the dual variable for the e constraint of (6). Then the dual of

(8) is

290 J. Ryan

Fig. 1.

maximize

wik$ c q.&+Uik+ c u,k+rj(o, e E 6(i) e 6 6(i)

ieV, k=l,..., m,

wik+wjk+uik+ujk+s,rO,

e=(i,j)EE, k=l,..., m,

Wik,Uik~O, ViE 1/, k=l,...,m,

&+,v&<O, VeeE, k= 1, . . . . m,

r,zO, Vig V,

~~20, VeeE.

(9)

(10)

(11)

(12)

Proposition 2. The value of (8) is bounded above by A(G) + 2.

Proof. By linear programming strong duality, the value of the program (8) is equal to the maximum value occuring in (9). Suppose we have a solution to the dual program.

If A(G) = 0, then the graph has no edges and clearly the value of (7), (and thus its relaxation (8)) is bounded above by A(G) + 1 = 1. So we assume that d(G) 2 1. As noted above, m has been chosen so that the problem (7) is feasible. Thus rnz A(G)+ 1~2.

Let IE{l,..., m> be chosen so that Ci~VWjk+Ce~E9ekiCjeVWil'CeeE4e/,

Vk= 1, . . . , m. Then, since the Wik’s and the qek’s are nonpositive,

Fractional total colouring

j, (i~~wi~+~~4P*)~m~i~~wit~~~qei)

s(A(G)+l) C wit+2 C qet* i6z V eeE

Using (11) and (12), for k=l,

i~vri+e~ESe5-i~vWil- C C 9ei-i~vUil- C C ueI is V eEd(i) ic V ecd(i)

291

(13)

- C lwit+ wj/)- C t”i/ + ujl)* e=(i, j)cE e=(i,j)EE

Since each edge has 2 endpoints, Ci, VCeEG(i) qe, is equal to 2CeEEqer, and

Ci,VC ecd(i) U,I is equal to 2CeEEue/. Also, since vertex i is in deg(i) edges,

-Ce=(ij)EE(~i,+ WJ) can be written -Ci,vdeg(i)wi~~-d(G)Ci~V~i,, and

- Ce_(i:j)EE(Uil+ Ujl) can be written - Ci, Vdeg(i)uill -d(G)Ci, vUi/.

Thus

-(d(G)+ 1) C uil-2 C ue/* ie V eeE

And so, using (13) and (14), and using (10) for k=I,

kE, i&wik+kt, e~Eqekfi~vri+e~Es~

<-(d(G)+ 1) iFvUi/-2 C u,~ ecE

(14)

=-i~vuil-e~E ue-r-~(G) C uil- C ue/ ie V f?CE

rl+d(G)+l

=d(G)+2. 0

Calling (8) the fractional total colouring problem, we have shown the following:

Theorem. Given a graph G, the value of the fractional total colouring problem for G lies between d(G) + 1 and d(G)+2.

We conclude by showing that, as is the case with the edge colouring problem, the

values of the integer programming problem and its linear programming relaxation

can differ by unity. Thus Behzad’s conjecture would not imply that the integer

rounding property (see e.g. [7]) holds for (7).

Let G be the 5-cycle, labeled as illustrated in Fig. 1. Then it is clear that 4=

,4(G) + 2 colours are needed to totally colour G. However the linear programming

relaxation (8) has the following solution of value 3. For k= 1,2,3,4, and for

292 J. Ryan

i = 1,2,3,4,5, let xjk = f. For k = 1,2,3,4, and for e = 1,2,3,4,5, let z& = f. Finally,

for k = 1,2,3,4, let y, = +. The constraints of (8) all are satisfied, and C”, = 1 yk = 3.

References

[I] M. Behzad, Graphs and their chromatic numbers, Doctoral Thesis, Michigan State University, East

Lansing, MI (1965).

[2] M. Behzad, The total chromatic number of a graph, in: A Survey of Combinatorial Mathematics

and its Applications, Proceedings Conf. Oxford 1969 (Academic Press, London, 1971) l-8.

[3] M. Behzad, G. Chartrand and J.K. Cooper Jr, The colour numbers of complete graphs, J. London

Math. Sot. 42 (1967) 225-228.

[4] J.A. Bondy and U.S.R. Murty, Graph Theory with Applications (North-Holland, New York, 1976).

[5] J.C. Meyer, Nombre chromatique total du joint d’un ensemble stable par un cycle, Discrete Math.

15 (1) (1976) 41-54.

[6] Rosenfeld, The total chromatic number of certain graphs, Notices Amer. Math. Sot. 15 (1968) 478.

[7] L.E. Trotter Jr, Discrete packing and covering: An annotated bibliography, in: M. O’hEigeartaigh,

J.K. Lenstra and A.H.G. Rinnooy Kan, eds., Combinatorial Optimization: Annotated Biblio-

graphies (Wiley, New York, 1987).

[8] N. Vijayaditya, On the total chromatic number of a graph, J. London Math. Sot. (2) 3 (1971)

405-408.