The NP-Completeness of Edge-Colouring

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The NP-Completeness of Edge-Colouring

Ian Holyer

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SIAM J. COMPUT, Vol. 10, No. 4, November 1981 (pp. 718-720) c

1981 Society for Industrial and Applied Mathematics 0097-5397/81/1004-0007 $01.00/0

Abstract. We show that it is NP-complete to determine the chromatic index of an arbitrary graph. The problem remains NP-complete even for cubic graphs.

Key words. computational complexity, NP-complete problems, chromatic index, edge-coloring

1. Introduction.

The chromatic index of a graph is the number of colors required to color the edges of the graph in such a way that no two adjacent edges have the same color.

By Vizing's theorem [1], the chromatic index is either d ord+ 1, where d is the maximum vertex degree.

We prove the conjecture (Garey and Johnson [2, p268]) that it is NP-complete to determine the chromatic index of an arbitrary graph. In fact, we prove the stronger result that it is NP-complete to determine whether the chromatic index of a cubic graph is 3 or 4. Thus this problem probably has no polynomial time algorithm.

The terminology and results of NP-completeness are given in [2]. It is clear that the chromatic index problem is in the class NP. To prove that the problem is NP-complete, we exhibit a polynomial reduction from the known NP-complete problem 3SAT which is de ned as follows. A set of clauses C = fC1;C2;:::;Crg in variables u1;u2;:::;us is given, each clause Ci consisting of three literals li;1;li;2;li;3, where a literal li;j is either a variable uk or its negation uk. The problem is to determine whetherC is satis able, that is, whether there is a truth assignment to the variables which simultaneously satis es all the clauses in C. A clause is satisifed if one or more of its literals has value \true".

2. The parity condition.

We will use the following lemma given in Isaacs [3].

LemmaLet G be a cubic, 3-edge-colored graph andV0 V(G) a set of vertices ofG. Let

E 0

E(G) be the set of edges of G which connect V0 to the remainder of the graph. If the number of edges of color i in E0 is ki (i= 1;2;3), then

k

1

k

2

k

3 (mod 2)

Proof. IfE12is the set of edges ofGwhich are colored with color 1 or 2, thenE12consists of a collection of cycles. Thus E12 meets E0 in an even number of edges, and so k1+k2 0 (mod 2) which gives k1 k2 (mod 2). Similarlyk2k3 (mod 2). 2

3. The components used in the construction.

Given an instanceC of the problem 3SAT, we will show how to construct a cubic graph Gwhich is 3-edge-colorable if and only if

C is satis able. The graphG will be put together from pieces or \components" which carry out speci c tasks. Information will be carried between components by pairs of edges. In a 3-edge-coloring ofG, such a pair of edges is said to represent the value T (\true") if the edges have the same color, and to representF (\false") if the edges have distinct colors.

Received by the editors January 31, 1980, and in nal form January 7, 1981

yUniversity Computer Laboratory, University of Cambridge, Cambridge, England. This work was supported by the British Science Research Council

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a

b

c

d e

Figure 1: The inverting component and its symbolic representation

The inverting component is shown with its symbol in Fig. 1. It was used by Loupekine (see [4]) to construct a large family of cubic graphs with chromatic index 4. Using the parity condition above, it may be checked that if this component is 3-edge-colored, one of the pairs of connecting edges marked a;b or c;d must have equal colors and the remaining 3 edges must have distinct colors. There is no further restriction on the possible colors of the ve connecting edges. Regarding the pair of edgesa;bas the input and the pairc;das the output, the component changes a representation of T to one of F and vice versa.

Figure 2: The variable-setting component made from 8 inverting components and having 4 output pairs of edges. More generally, it is made from 2n inverting components and has n output pairs.

The truth or falsity of each variableui will be represented by a variable-setting component such as that shown in Fig. 2. The component shown has 4 pairs of output edges, but in general the component representing ui should have as many output pairs as there are appearances of ui or ui among the clauses of C. It may be checked that in any 3-edge-coloring of a variable-setting component, all the output pairs must represent the same value.

The truth of each clausecj will be tested by a satisfaction-testing component as shown in Fig. 3. This component can be 3-edge-colored if and only if the three input pairs of edges do not all represent F. The remaining connecting edges will be discussed later.

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Figure 3: The satisfaction testing component.

4. The main theorem.

We are now in a position to prove the following theorem.

Theorem. It is NP-complete to determine whether the chromatic index of a cubic graph is 3 or 4.

Proof. The problem is clearly in the class NP. We exhibit a polynomial reduction from the problem 3SAT. Consider an instance C of 3SAT and construct from it a graphG as follows.

For each variable ui take a variable-setting component Ui with one output pair of edges associated with each appearance of ui orui among the clauses ofC. Take also a satisfaction- testing component Cj for each clause cj. Suppose literal lj;k in clause cj is the variable ui. Then identify the kth input pair of Cj with the associated output pair of Ui. If, on the other hand, lj;k is ui, then insert an inverting component between the kth input pair of Cj and the associated output pair of Ui. The resulting graph H still has some connecting edges unaccounted for. The cubic graph G is formed from two copies of H by identifying the remaining connected edges in corresponding pairs.

The graphGhas a 3-edge-coloring if and only if the collection C of clauses is satis able, as can be veri ed using the properties of the components developed above. Moreover, the graph G can be produced fromC using a polynomial time algorithm, so we have the result.

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5. Comments.

The above theorem may give some insight into the diculty in classifying graphs according to their chromatic index. At any rate, it probably excludes the possibility of a polynomially checkable criterion, and it indicates that the restriction to cubic graphs is no easier.

Acknowledgements.

I would like to thank M. Garey and D. Johnson for suggesting a simpli cation in the proof.

REFERENCES

[1] S. Fiorini and R,J. Wilson,Edge-colourings of Graphs, Pitman, London, 1977.

[2] M.R. Garey and D.S. Johnson, Computers and Intractability, W.H. Freeman, San Francisco, 1979.

[3] R. Isaacs, In nite families of non-trivial trivalent graphs which are not Tait colorable, Amer. Math. Monthly, 82 (1975), pp. 221-239.

[4] ||{, Loupekine's snarks: A bifamily of non-Tait-colorable graphs, unpublished.

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