Acyclic Graph Coloring and the Complexity of the Star Chromatic Number

David R. Guichard WHITMAN COLLEGE

WALLA WALLA, WASHINGTON

ABSTRACT

Star chromatic number, introduced by A. Vince, is a natural gener- alization of chromatic number. We consider the question, “When is x* < x?“ We show that x* < ,y if and only if a particular digraph is acyclic and that the decisioin problem associated with this question is probably not in NP though it is both NP-hard and NP-easy. 0 1993 John Wiley & Sons, Inc.

1. INTRODUCTION

Vince [6] introduced a natural generalization of the chromatic number, the star chromatic number x*. He proved, among other things, that x ( G ) - 1 < ,y*(G) 5 x ( G ) , that , ~ * ( C Z ~ + ~ ) = 2 + (l /n), and that ,y*(K,) = n. In some sense a graph G for which ,y*(G) < x ( G ) is easier to color than one for which these two numbers are the same. Vince closed his paper with four questions, the most general being “What determines whether x* = x?” Here we provide a useful and interesting characterization of ,y*(G) < q E Q, and show that the question “is x*(G) = x(G)?” is in one sense precisely as hard as the NP-complete problems but probably is not in NP.

2. THE STAR CHROMATIC NUMBER

First, a basic definition:

Definition. lxlk is the distance from x to the nearest multiple of k .

Journal of Graph Theory, Vol. 17, No. 2, 129-134 (1993) 0 1993 John Wiley & Sons, Inc.

For any real number x and positive integer k the circular norm

CCC 0364-9024/93/020129-06

130 JOURNAL OF GRAPH THEORY

The following was articulated in Bondy and Hell [2] but was implicit in [6]:

Definition. Let k and d be positive integers. A (k,d)-coloring of a graph G is a function c : V(G) - Zk such that for any adjacent vertices u and U, I c ( u ) - c ( u ) I ~ 2 d.

An ordinary (proper) coloring of a graph is a ( k , 1)-coloring, and any such coloring of a graph immediately provides (ik, i)-colorings for any positive integer i. Vince focused his attention on the ratio k / d and observed that this ratio can sometimes be made smaller than x(G). He defined and studied a new parameter:

Definition. The star chromatic number ,y*(G) is inf{k/dIG has a ( k , d)- coIoring}.

(The form of Vince’s definition was slightly different than this; the definition here is from [2]). One of his major results was

Theorem. If G is a graph on n vertices, then

x* = min {k/dlG has a ( k , d)-coloring and k 5 n}.

Thus the star chromatic number is always rational and always attained by some ( k , d)-coloring.

It is useful to consider a more general class of graph “colorings,” namely, functions from the vertices of a graph G to the interval I = [0, 1). A “good” coloring is one for which the colors of adjacent vertices are far apart, and a best coloring is one for which the colors are as far apart as possible. More precisely, for a coloring c : V - I we let d(c ) = min, adj Ic(u) - c(u)I1, and XG = supd(c), taking the sup over all functions c : V(G) - I . It is easy to see that x * ( G ) = ~ / X G .

Vince’s proof of the theorem above proceeded by considering these more general real valued colorings, and using methods and results from continuous mathematics. Bondy and Hell [2] supplied a purely combinatorial proof that is somewhat shorter than Vince’s, but we find it convenient to consider colorings c : V - I in some circumstances.

3. ACYCLIC COLORINGS

We establish a condition that is equivalent to ,y*(G) 5 q for a graph G , and we then prove that the question “Is x* = x?” is NP-equivalent, that is, it is in some sense exactly as hard as the NP-complete problems. First we need one more definition.

Definition. Given a k-coloring c : V(G) - 4 of a graph G and d I min, adj Ic(u) - c(u)lk, the digraph Hk,d(c) has the same vertex set as

TUTTE POLYNOMIALS FOR TREES 131

G and an edge directed from u to w if u is adjacent to w in G and c(w) = c(u) + d (mod k ) .

This operation Hk,d(c) appears in Vince’s paper in slightly different form. We write H ( c ) when k and d are clear from context. At least one direction of the following theorem is implicit in Vince’s work:

Theorem 1. coloring c of G.

,y*(G) < k / d if and only if H(c) is acyclic for some ( k , d ) -

Proof. This first part of the proof appears in Vince [6] in a different context. Suppose H ( c ) is acyclic. Let c’(u) = c(u) /k and number of vertices u1, u2,. . . , u, so that if (u; , uj) is an edge in H ( c ) then i > j . For rational E > 0 we define f ( u i ) = c‘(ui) + c / i and note that for E sufficiently small

an edge of H and likewise If(ui) - f (u j ) l l > d / k if (ui, u j ) is not an edge. Hence ,y*(G) < k / d .

Now suppose ,y*(G) < k / d and let c1 : V ( G ) -, I be a coloring of G such that d(c1) > d / k . Define CO(U) = Lkcl(u)l/k; kco(u) is a ( k , d ) - coloring of G . Let E(U) = cl(u) - co(u) 2 0; note that E ( U ) < l / k . Num- ber the vertices u1, u2, ..., u, so that ~ ( u i ) 2 E ( u ~ + I ) .

f(vi) E [O, I), If(vi) - f(vj)Il > Ic’(vi) - c’(vj))l = d / k if (vi, ~ j ) is

Suppose there is an edge in H(kco) from ui to uj with i < j . Then

If C O ( U ~ ) = CO(U~) + d / k then 0 < C l ( V j ) - CI(U~) 5 C o ( V j ) - C O ( U ~ ) = d / k so d ( c l ) 5 d / k , a contradiction. Hence c0(uj) = 0 and c0(ui) = ( k - d ) / k , so -1 < c l ( u j ) - c l ( u i ) 5 - ( k - d ) / k and again d ( c l ) 5 d / k , a contradiction. Thus, all edges (ui, u j ) have i Z j so H(kc0) is acyclic. I

We will call a k-coloring c of G an acyclic coloring if Hk, (c) is acyclic (and more generally, we could refer to an acyclic ( k , d)-coloring, though we will not need this notion).

Colollary. ,y*(G) < k if and only if G has an acyclic k-coloring.

The following proposition was proved independently by Zhu [7].

Proposition. then ,y*(G) = ,y(G).

If the graph G has a vertex u adjacent to all other vertices

Proof. Let c be a ,y(G)-coloring of G ; without loss of generality suppose that c(u) = ,y(G) - 1. Orient the graph G - u by directing an edge from u to w if c(u) < c(w) . It is known (see [l], for example) that this digraph

132 JOURNAL OF GRAPH THEORY

contains a directed path on ,y(G) - 1 vertices, which must therefore be colored 0, 1 , 2, . . . , ,y(G) - 2 in that order. Thus, this path is also contained in the graph Hk, ( c ) and forms a cycle with vertex u. Since G has no acyclic ,y(G)-coloring, ,y*(G) = ,y(G). I

4. COMPLEXITY

Garey and Johnson [3] use the term NP-equivalent to describe problems that are Turing reducible to an NP-complete problem and to which some NP-complete problem is Turing reducible. An NP-equivalent problem has a polynomial time solution if and only if P = NP. See Garey and Johnson, especially chapter 5 , for definitions and discussion of oracle Turing machines and Turing reducibility.

We denote by 3COLOR the problem of determining whether a graph is 3 colorable, and by GRAPH HOMOMORPHISM the problem of determining whether two graphs homomorphic; both are NP-complete. Denote by STAR- COLOR the problem of determining for a graph G whether x*(G) < x ( G ) . Note that a (k, d)-coloring of G is a homomorphism from G to the graph Gf with vertex set Zk and edge set {ij((i - j l k L d}. Indeed,as Bondy and Hell remark, ,y*(G) I k/d if and only if G is homomorphic to Gf.

Theorem 2. STARCOLOR is NP-equivalent.

Proof. We show first that STARCOLOR is Turing reducible to GRAPH HOMOMORPHISM. With at most IV(G)l calls to the oracle we can determine x(G). Then x*(G) < ,y(G) if and only if ,y*(G) 5 k’/d’, where

Finally, ,y*(G) 5 k‘/d‘ if and only if G is homomorphic to G$, which we can determine with one more call to the oracle.

Now we show that 3COLOR is Turing reducible to STARCOLOR. We are given a graph G with x(G) 5. 3. Ask the oracle whether ,y*(G) < ,y(G). If so, let G’ = G U K3 and ask whether ,y*(G’) < ,y(G’). If so, x ( G ) 2 4 else x ( G ) = 3. If ,y*(G) = ,y(G), let G’ = G U H , where 4 = , y (H) > ,y*(H). Then ,y(G) = 3 if and only if ,y*(G’) < x(G’). I

An obvious remaining question is whether STARCOLOR is in NP. The answer is probably not. That is, we show that if STARCOLOR is in either NP or co-NP then NP = co-NP. The key is the following theorem of Leggett [5] as stated in [3]:

TUlTE POLYNOMIALS FOR TREES 133

Theorem. complete, Ly m F p LO and Lz m F p L& then

If L 1 , L l , and L2 are languages such that L1 and L2 are NP-

Lo E NP U CO-NP NP = CO-NP.

Here L“ denotes the complement of L and x,“’ denotes “nondeterministic polynomial time conjunctive truth-table reducibility” as described in Ladner, Lynch, and Selman [4]. Informally, A mYPB if there is a polynomial time NOTM (nondeterministic oracle Turing machine), restricted to halt in the “no” state whenever the oracle says “no,” which answers question A using an oracle for question B.

Theorem 3. The hypotheses of Leggett’s theorem are satisfied if L1 and L2 are taken to be 3COLOR and LO is STARCOLOR.

Proof. First, we show that (3COLOR) is reducible to (STARCOLOR)’. Let G’ = G + u (i.e., add a new vertex connected to all vertices of G ) and G” = G’ U H , where 5 = x ( H ) > ,y*(H). By the proposition above, x*(G’) = x(G’), so x ( G ) 2 4 if and only if ,y(G’) 2 5 if and only if x*(G”) = ,y(G”). Note that this reduction is actually a garden variety many-one reduction.

Next we show that (3COLOR)’ is reducible to (STARCOLOR). Suppose that ,y*(G) < ,y(G). Then x ( G ) 2 4 if and only if ,y*(G) < x ( G ) and x*(G U K3) < ,y(G U K3). (Note that the TM is not permitted to test the condition x* < X-it must correctly guess that this is true and then verify it. Actually, the procedure described below will work if x* < x, but is unnecessarily complicated.)

If ,y*(G) = x ( G ) the NOTM will “guess” a graph G’ such that x(G’) = x ( G ) = k and ,y*(G’) < ,y(G’). We need to establish the existence of such a graph G’ that is such that a TM can verify the pertinent properties of G’ in polynomial time.

We begin with a graph Go, an edge critical subgraph of G with the same chromatic number. GO - (u ,u) can be (k - 1)-colored, necessarily giving vertices u and u the same color (say 0), and Go can then be k-colored by giving u color k ; fix such a coloring of Go - ( u , v ) . Recall the Hajos operation: if (u1, w l ) and (v2, w2) are edges in the vertex disjoint graphs H1 and Hz, form a new graph V(H1, v1, w l , H2 , v2, w2) by removing both edges, joining w1 to w2 by an edge, and identifying u1 and u2. It is easy to see

Now, in Go the vertex u is necessarily adjacent to vertices colored 0, say wl , . . . , wr- Let Gi = V(Gi-l, wit v,Go, wi, u ) for i = 1, . . . , r . (Note the abuse of notation-we work with disjoint copies of the graphs Gi-l and Go, so the w i and u represent four distinct vertices in this expression, and moreover by w j and u in Gi we mean the copies of w j and u in Gi-1.) Let G’ = G,. We think of G’ as a “base copy” of Go with each edge (v, w i )

that if x(H1) = x ( H 2 ) then x(V(HI , v1, wl,H2, v2, W Z ) ) = x ( H 1 ) .

134 JOURNAL OF GRAPH THEORY

Figure 1

replaced by a copy of Go - (u , wi) plus an extra edge joining u to its copy, as suggested by Figure 1.

G’ can be k-colored using the coloring fixed above on each copy of Go (with edges removed) but coloring u with color k. This coloring of G’ is acyclic, since a cycle would have to proceed from u (color k ) to a copy of u (color 0) and then back to u without using any other vertex colored O-but any such cycle must use at least one wi.

Now our NOTM guesses Go, G’, and explicit information showing how G’ is constructed and how Go sits in G . This allows the TM to verify in polynomial time that x ( G ) L ,y(Go) = ,y(G’). Using the oracle the TM verifies that x*(G’) < ,y(G’), and finally uses the fact that ,y(G’) 2 4 if and only if x*(G’ U K3) < x(G’ U K3). I

References

[l] B. BollobBs, Extremal Graph Theory. Academic Press, London (1978). [2] J. A. Bondy and P. Hell, A note on the star chromatic number. J . Graph

Theory 14 (1990) 479-482. [ 3 ] M. R. Garey and D. S. Johnson, Computers and Intractability. W. H.

Freeman, San Francisco (1979). [4] R. E. Ladner, N. A. Lynch, and A. L. Selman, A comparison of polyno-

mial time reducibilities. Theoret. Comput. Sci. 1 (1975) 103- 123. [5] E. W. Leggett, Jr., Tools and Techniques for Classifiting NP-Hard Prob-

lems. Doctoral Thesis, Department of Computer and Information Sci- ences, Ohio State University (1977).

[6] A. Vince, Star chromatic number. J . Graph Theory 12 (1988) 551-559. [7] X. Zhu, Star chromatic number and products of graphs. J . Graph Theory

16 (1992) 557-570.