A Note on the Star Chromatic Number

J.A. Bondy

Pavol Hell UNlVERSlN O f WATERLOO

SIMON FRASER UNWERSIN

ABSTRACT

A. Vince introduced a natural generalization of graph coloring and proved some basic facts, revealing it to be a concept of interest. His work relies on continuous methods. In this note we make some simple observations that lead to a purely combinatorial treatment. Our methods yield shorter proofs and offer further insight.

The following notion is implicit in [3].

Definition 1. Let k and d be positive integers such that k 2 2d. A (k,d)- coloring of a graph G = ( V , E ) is a mapping c: V-* Zk such that, for each edge uu E E, (c(u) - c(V)lk 2 d, where [xik := min(lx1, k - 1x1).

Remark 1. A (k, 1)-coloring of G is simply a proper k-coloring of G ; there- fore the chromatic number x ( G ) is the smallest k for which G admits a (k, 1)- coloring.

Proposition 1. If G has a (k, d)-coloring and k/d I k ’ /d ’ , where k ‘ and d ’ are positive integers, then G has a ( k ’ , d ‘)-coloring.

Proof. Let c: V + Zk be a (k, d)-coloring of G = (V, E). Define the map- ping c’: V + &. by

c’(u) = -c(u) , for all u E V L ‘ J Consider an edge uu, and assume that c(u) > c(u). Since c is a (k, d)-coloring of G ,

d 5 C(U) - C(U) 5 k - d .

Journal of Graph Theory, Vol. 14, No. 4, 479-482 (1990) 0 1990 by John Wiley & Sons, Inc. CCC 0364-9024/90/040479-04$04.00

480 JOURNAL OF GRAPH THEORY

Therefore

1 c’(u) + d’ = (c(u) + d)] 5 c’(u) 5 (c(u) + k - d )

i c’(u) + k ’ - d’,

and

d ’ 5 c’(u) - c’(u) 5 k’ - d‘.

Thus c’ is a (k’,d’)-coloring of G.

Corollary 1. k’fd‘ = kfd and gcd(k’,d’) = 1.

If G has a (k,d)-coloring, then it has a (k’,d’)-coloring with

Proposition 2. If G is a graph on n vertices that has a (k,d)-coloring with gcd(k,d) = 1 and k > n, then G has a (k’,d’)-coloring with k’ < k and k‘fd’ < k/d.

Proof. There is an integer i that is not the color of any vertex of G ; without loss of generality assume that i = d. If there are vertices of color 26, we re- color them with color 2d - 1; the result is still a (k, d)-coloring of G. Continu- ing in this fashion, recoloring all vertices of colors 3d, 4d, . . . , ad, where a is the smallest integer such that a d = 1 (in Z k ) , we arrive at a (k,d)-coloring of G in which no vertex receives a color from the set S := {d, 26, . . . ,ad}. That such an a exists follows from the assumption that gcd(k, d ) = 1. Define k ’ = k - a. Since the colors in the set S are not used, we may rename the other colors, changing color x to x - [ { y E S: y < x}l , thereby obtaining a mapping c’: V + Z,. We shall show that C’ is a coloring of the desired type. Define p := (ad - l)/k, and denote by Ij the interval { j, j + 1, . . . , j + d - 1) of Z, (not of Zr). Observe that each interval I,, j # 1, contains exactly /3 elements of S, while I , contains exactly /3 + 1 elements of S. Noting that both ends of the interval I, belong to S, it is easy to see that c’ is a (k’,d’)- coloring of G, where d ’ := d - p. Moreover,

k’ k - a k(k - a) k <- -=-= d ’ d - P d ( k - Q ) + l d ’

as required. The following parameter is the main object of study in [3]:

Definition 2. The star chromatic number x*(G) of a graph G is defined by

x*(G) := inf{k/d: G has a (k , d) - coloring}.

NOTE ON THE STAR CHROMATIC NUMBER 481

Corollary 2. If G is a graph on n vertices, then

x*(G) = min{k/d: G has a ( k , d) - coloring and k I n} ,

Proof. By Corollary 1 and Proposition 2, if G has a (k, d)-coloring then it has a (k’, d’)-coloring with k’ 5 n and k ’ / d ’ 5 k/d. Therefore

x*(G) = inf{k/d: G has a (k, d ) - coloring and k I n}.

Because the set { k / d : G has a (k,d) - coloring and k i n} is finite, the infi- mum can now be replaced by a minimum.

Remark 2. Corollary 2 is, in essence, Theorem 3 of [3]. It implies that x* is rational; moreover, by Proposition 1, for every rational number r 2 x* and every pair of positive integers k and d with r = k/d, there exists a (k,d)- coloring. In particular, G has a (k, d)-coloring if and only ifx*(G) 5 k/d.

Remark 3. By Remark 1, x* 5 x. On the other hand, x* > x - 1, because otherwise Corollary 2 would imply the existence of a (k, d)-coloring with k/d I x - 1, and hence, by Proposition 1, a (x - 1 , 1)-coloring. Thus, as shown in [3],

Definition 3. A homomorphism f: G + H , for graphs G and H, is a mapping f: V ( G ) + V ( H ) such thatf(u)f(u) E E ( H ) whenever uu E E(G).

Most generalizations of graph coloring may be expressed in terms of graph homomorphisms. In our case, a (k,d)-coloring c of G is simply a homomor- phism from G to the graph G;‘ whose vertex set is Z, and whose edge set is { i j : ( i - j ( , 2 d} (the complement of the (d - I)-st power of the k-cycle).

Proposition 3. NP-complete to decide whether x* I k/d.

For any fixed positive integers k and d such that k > 2d, it is

Proof. In [2] it is shown that, for a fixed nonbipartite graph H , it is NP- complete to decide whether there is a homomorphism f: G + H . Since G: is easily seen to be nonbipartite when k > 2d (this is also implied by Corollary 3 below), we deduce that it is NP-complete to decide whether a graph admits a (k, d)-coloring. The result follows by Remark 2.

Remark 4. For any fixed positive integers k ‘ and d‘ such that k’ > 2d’, it is also NP-complete to decide whether x* < k ’ / d ‘ . To see this, note that, by Corollary 2, x*(G) 5 k/d if and only if x*(G) < k’/d’ , where k’/d’ = min{k”/d”: k f f / d f f > k/d and k“ 5 n} and n is the number of vertices of G ; thus k‘, d‘, can be found in time polynomial in the length of the input, G.

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In [3] it is also proved that x * ( G i ) = k / d . This is easy to deduce from a result of Albertson and Collins [ I ] . For completeness, we prove a slight gener- alization of their result and show how to apply it to derive the above fact. For graphs G and K , let u ( G ) denote the number of vertices of G, and u ( G , K ) the maximum number of vertices in a subgraph of G that admits a homomorphism to K .

Proposition 4. there is a homomorphismf: G-, H, then

Let G , H and K be graphs, where H is vertex-transitive. If

Proof. Let H,, H,, . . . , H, be the largest subgraphs of H that admit homo- morphisms to K . Since H is vertex-transitive, each vertex of H is covered by the same number, say p , of the graphs HI. It follows that q * u(H, K ) = p . u ( H ) . Let G, be the subgraph of G induced byf-'(V(H,)). Then each vertex of G belongs to p of the graphs G I , and each G, admits a homomorphism to K . Therefore q * u(G, K ) 2 p 1 u(G ).

Corollary 3. x * ( G f ) = k / d .

Proof. Since Gi has a ( k , d)-coloring, it remains to prove that there is no homomorphism G f + Gf: with k' /d ' < k / d . This is guaranteed by Proposi- tion 4, with G:= G f , H: = Gf: , and K : = K , ; it suffices to observe that each G: is vertex-transitive and to check that the size of a largest independent set in G: is y, that is, u(GZ, K ) = y .

Remark 5. If there is a homomorphism f : G + H, then any ( k , d)-coloring of H (which is itself a homomorphism of H to G i ) can be composed with f to obtain a (k, d)-coloring of G ; hence

Thus, if there are homomorphisms G + H and H + G, then x * ( G ) = x * ( H ) . For instance, as observed in [3], if G has a proper n-coloring and a largest clique K , , then x * ( G ) = x * ( K , ) = n . (The second equality follows from Corollary 3 and the fact that K, = G,!.)

References

[ 1 1 M . 0. Albertson and K. L. Collins, Homomorphisms of 3-chromatic

[2] P. Hell and J . NeSetiil, On the complexity of H-colouring. J . Combinat.

[3] A. Vince, Star chromatic number. J . Graph Theory U (1988) 551-559.

graphs. Discrete Math. 54 (1985) 127- 1 .

Theory Ser. B , 48 (1990) 92- 110.