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  • DISCRETE MATHEMATICS

    ELSEVIER Discrete Mathematics 191 (1998) 139 148

    Complexity of choosing subsets from color sets 1

    J. Kratochvf l a, Zs. Tuza b, M. Voigt c'*

    a Department of Applied Mathematics, Charles University, Prague, Czech Republic b Computer and Automation Institute, Hungarian Academy of Sciences, Budapest, Hungary

    c Institut fiir Mathematik, Technische Universitdt llmenau, Postfaeh 0565 98684, Ilmenau, German),

    Received 11 June 1996; revised 11 March 1997; accepted 23 December 1997

    Abstract

    We raise and investigate the algorithmic complexity of the following problem. Given a graph G = (V,E) and p-element sets L(v) for its vertices v C V such that IL(u)t3 L(v)l >1 p + r for all edges uv C E, do there exist q-element subsets C(v)C L(v) with C(u)A C(v)= 0 for all uv E E? Here p,q, r are positive integers, p ~>q and p + r >~2q. We characterize precisely which triples (p, q, r) admit a polynomial solution, and for which ones the problem is NP-complete. Moreover, it is shown that for some restricted ranges of p and r with respect to q, the existence of subsets C(v) C L(v) for every collection {Z(v) lv ~ V}, is closely related to Turfin's problem on uniform hypergraphs. @ 1998 Elsevier Science B.V. All rights reserved

    Keywords: List colorings; Set choosability; Complexity; Satisfiability

    1. Introduction

    Let p,q , r be natural numbers, and G=(V,E) a graph with vertex set V and edge set E. A (p,r)-assignment of G is a col lection

    Lf = {L(v) ] v ~ V}

    of lists (color sets) assigned to the vertices such that

    [L(v)l = p for all v C V,

    and

    IL(u) U L(v)] >>. p + r for all uv E E.

    l Parts of the research were done during visits of various subsets of the authors to Technical University Ilmenau and to Charles University Prague. * Corresponding author. E-mail: voigt@mathematik.tu-ilmenau.de.

    0012-365X/98/$19.00 Copyright @ 1998 Elsevier Science B.V. All rights reserved PH S001 2-365X(98)00 1 01-0

  • 140 .L Kratochvil et al./Discrete Mathematics 191 (1998) 13~148

    An ~q~-admissible q-set colorin9 of G is a collection

    iq or q + 1 >ip>iq>>-r should hold (Theorem 2.2).

    Section 3 presents a partial result on (p, q, r)-choosability, giving a sufficient condi- tion for its solvability in linear time (Theorem 3.1). In that case, the close relationship between q-set choosability and a Tur~in-type function in extremal hypergraph theory is pointed out. One of the consequences is that, for some values of p, q, r, choosability is much easier to decide than colorability with respect to a given list assignment.

    2. The complexity of (p,q,r)-list colorings

    In this section we study the following decision problem parametrized by (fixed) integers p, q, r:

  • J. Kratochvll et al./Discrete Mathematics 191 (1998) 139-148 141

    00000000000~ 00000000000 0000000000 0 0 0 0 ~ 00000 00000000 0000000 0000

    1 ~"

    Fig. I. Complexity of (p,q,r)-list coloring - = polynomial, o - NP-complete. (Illustration for q = 6.)

    (p, q, r)-iist coloring ((p, q, r)-LC) Instance: A graph G and a collection of lists, Lf = {L(u) u E V(G)}, such that

    (i) [L(u)l = p for every u c V(G), and (ii) [L(u) U L(v)[ >~ p + r for every edge uv C E(G).

    Question: Is there an LP-admissible q-set coloring of G, 1.e., a collection c~= {C(u)l u E V(G)} such that

    (i) C(u)C_ L(u) for every u, (ii) [C(u)[ = q for every u, and (iii) C(u) A C(v) = 0 for every edge uv E E(G)? Note that the question only makes sense for p>~q (if p~r (if p~ max {q + 2,r + 1} and

    (ii) solvable in linear time for p = r>~q and for q

  • 142 J. Kratochvll et al. ID&crete Mathematics 191 (1998) 139-148

    easy, since C(u)=L(u) must hold then for every u E V(G), and therefore the instance is feasible if and only if the lists of admissible colors of adjacent vertices are disjoint.

    The only interesting case is p = q + 1, r = 0. (Note that once we prove this particular case, part (ii) of the theorem will be proved via Lemma 2.1).

    Given an instance of (q+ 1, q, 0)-LC, we introduce p I V(G)I boolean variables x(u, a) (uE V(G), aEL(u)). We will encode the boolean values so that x(u,a)=true iff a E C(u), and construct a formula so that is satisfiable if and only if (G, ~) admits a q-set coloring. For every edge uv E E(G) and every color a E L(u)AL(v), we set ~u,v,a=-nx(u,a) V-~x(v,a). For every uc V(G) and every two colors a, bCL(u), we set ~u,~,b =x(u, a)Vx(u, b). The subformulas ..... control the disjointness of C(u) and C(v), while the subformulas 45 .... b guarantee that L(u) - C(u) has at most one element. Therefore, the 2-formula

    ~'= A ~,~,~ A A a~ .... b uvCE(G),aEL(u)nL(v) uEV(G),a,b@L(u)

    is satisfiable if and only (G, ,e) admits a q-set coloring. Since the length of q~ is linear in E, the assertion (ii) follows by the well known fact that 2-SATISFIABILITY is solvable in linear time. []

    The NP-completeness part of Theorem 2.2 will be proved via an auxiliary problem called (2.5, 1, 1.5)-list coloring: (2.5, 1, 1.5)-list coloring ((2.5, 1, 1.5)-LC) Instance: A graph G and a collection of lists, ~,e = {L(u) [ u E V(G)}, such that

    (i) 2~

  • J. Kratochvil et al./Discrete Mathematics 191 (1998) 139 148 143

    Obviously, (G, ~) is an instance of (2.5, 1, 1.5)-LC, and it is feasible if and only if ~b is satisfiable. (The truth valuation to be taken is x=true i f fx is colored by {x}.)

    A graph G(u,b) equipped with a (p,r)-assignment, and with a specified vertex u and a color b E L(u), is called a ( p, q, r )-chooser if it is a legal and admissible instance of (p ,q , r ) -LC and every admissible q-set coloring (g satisfies bE C(u).

    Lemma 2.4. I f p >~ q + 2, p >~ r + l, and i f a (p, q, r)-chooser exists, then (2.5, 1, 1.5)- LC ~ (p ,q , r ) -LC .

    Proof. Let (G, LP) be an instance of (2.5, 1,1.5)-list color. We build a graph G' equipped with lists U by a series of local adjustments. If a vertex ui E V(G) is asso- ciated with a list of size s=s( i ) , we consider p- s(i) new colors bi, l,bi,2 . . . . . bi.q i, Fi. 1, ri,2 . . . . . ?'i, p-q s ( i )+ l and p - q - s(i) + 1 disjoint copies of the chooser G(ui./, ri./), j = 1,2 . . . . . p - q - s(i) + 1. The new list for ui will be U(u i ) =L(u i )U {bi.l,b,,2 . . . . . bi, q -1 , ri, l,?'i,2 . . . . . ri, p -q s ( i )+ l} , and ui will be adjacent to ui, j, j = 1,2 . . . . . lgp_q s'(t)--1, This construction is performed for every vertex ui ~ V(G), where the color sets

    {bi, l, bi.2 . . . . . bi,q-l,ri.|,ri.2 . . . . . ri, p-q-.~(i)+l} and {bk,~,bk.2 . . . . . bLq- l , rL i , rk.2 . . . . . r~,p_,t_,~!+l } are disjoint for each pair ui, uk of adjacent vertices. The pair (G ' ,S ' ) obtained in this way is an instance of (p ,q , r ) -LC . In any admissible q-set color- ing ~' of G t, r,.j E C(ui, j ) ( j = 1,2 . . . . . p - q - s( i) + 1), and hence C(ui) C L(ui) U {bi, l,bi,2 . . . . . bi, q - l} for every ui E V(G) . It follows that IC(ui) N L(ui)[ >~ 1 for ever)' ui, and hence c~, induces a proper coloring of G. The other direction (i.e., that (G', Y ' ) is feasible if (G, Lt') is feasible) is trivial. []

    Lemma 2,5. A (p, q, p - 1)-chooser exists fo r ever), p >q >~ 1.

    Proof, Below, i , j , k denote integers and h=(h l ,h 2 . . . . . hp) is a vector of length p whose coordinates take values from {q + 1,q + 2 . . . . . p}. The graph G(u,b) will have the vertex set

    V(G(u ,b ) ) ={u} U {ui, i I q + l

  • 144 J. Kratochvil et al./Discrete Mathematics 191 (1998) 139-148

    Note that elements denoted differently are different. The reader may easily verify that G(u,b) with the constructed list assignment is an instance of the (p,q, p - 1)-LC problem.

    The lists admit a proper q-set coloring; e.g., the subsets

    C(u) : {b l ,b2 , . . . ,bq} ,

    C(ui, j ) :{bi , b(i,j, 2) ..... b(i,j,q)}, q+ l3r+p, or (2) 2r2p+r,

    then the path P4 is not (p,q,r)-choosable.

  • .L Kratochvil et al./Discrete Mathematics 191 (1998) 139-148 145

    ProoL Denote the vertices of P4 by Vl, /)2, /)3, v4 (in this order along the path). We construct two types of list assignments, depending on the relationship between p and r. In each case, the lists will be intervals. Let L~ denote the list assigned to the vertex v,.

    Case 1. 2r >~ p and 4q>3r+p Define

    c, -- {1 . . . . . p},

    L2 = {r+l . . . . . r+p}, Ls - -{2r+l . . . . . 2 r+p}, L4 - - {3r + 1 . . . . . 3r + p}.

    Notice that IL, I = p and ILi UL/I = p r for all i , j E {1 . . . . . 4} and i = j + 1. Suppose Ci has been selected from Li and ICi[ = q. We consider the interval I = {p+ 1 . . . . . 3r}.

    First, we ask about the minimum number of elements of C2 included in this interval. Since 2r ~> p, no element of L2 (and hence none of C2 either) is greater than 3r Furthermore, there are (p + 1) - (r + 1) elements of L2 smaller than p + 1. On the other hand, we consider L1 and obtain that at least q - r elements of the interval {r + 1 . . . . . p} belong to C1. Thus, at most (p + 1 ) - (r + 1 ) - (q - r) = p - q elements of C2 are smaller than p + 1, and therefore IC2 N II >~q - (p - q) =2q - p.

    Analogous investigations lead to the inequality IC3 n i I >~2q-p. Since C2 and C3 arc disjoint, we obtain 4q- 2p ~< Ill = 3r - p, a contradiction to the assumption 4q > 3r+p.

    Case 2. 2r2p+r Define

    L, = {1 . . . . . p},

    L2 -- { r+ 1 . . . . . r+p}, L3 ={p+l . . . . . 2p}, L4 = {r + p + 1 . . . . . r + 2p}.

    It is easy to check that ILl I= p and ILl UL j IF p + r for all i , j C {1 . . . . . 4} and i= j + 1. Supp

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