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Nordhaus-Gaddum-Type Theorems for Decompositions Into Many Parts Nordhaus–Gaddum- Type Theorems for Decompositions into Many Parts Zoltan Fu¨ redi,1,2 Alexandr V. Kostochka,1,3 Riste Sˇ krekovski,4,5 Michael Stiebitz,6 and Douglas B. West1 1UNIVERSITY OF ILLINOIS, URBANA, ILLINOIS 61801 E-mail: [email protected]; [email protected]; [email protected] 2RE´ NYI INSTITUTE OF MATHEMATICS, P.O.BOX 127, BUDAPEST 1364, HUNGARY 3INSTITUTE OF MATHEMATICS, 630090 NOVOSIBIRSK, RUSSIA 4CHARLES UNIVERSITY, MALOSTRANSKE´ NA´M. 2/25 118 00, PRAGUE, CZECH REPUBLIC 5UNIVERSITY OF LJUBLJANA, JADRANSKA 19 1111 LJUBLJANA, SLOVENIA E-mail: [email protected] 6TECHNISCHE UNIVERSITA¨ T ILMENAU D-98693 ILMENAU, GERMANY E-mail: [email protected] Received February 17, 2003; Revised April 4, 2005 —————————————————— Contract grant sponsor: Hungarian National Science Foundation; Contract grant number: OTKA T 032452 (to Z.F.); Contract grant sponsor: NSF; Contract grant number: DMS-0140692 (to Z.F.), DMS-0099608 (to A.V.K. and D.B.W.), DMS- 0400498 (A.V.K. and D.B.W.); Contract grant sponsor: NSA; Contract grant number: MDA904-03-1-0037 (D.B.W.); Contract grant sponsor: Dutch-Russian; Contract grant number: NOW-047-008-006 (A.V.K.); Contract grant sponsor: Ministry of Science and Technology of Slovenia; Contract grant number: Z1-3129 (R.S.); Contract grant sponsor: Czech Ministry of Education; Contract grant number: LN00A056 (to R.S.); Contract grant sponsor: DLR; Contract grant number: SVN 99/003 (to R.S.). ß 2005 Wiley Periodicals, Inc. 273 274 JOURNAL OF GRAPH THEORY Published online 8 August 2005 in Wiley InterScience(www.interscience.wiley.com). DOI 10.1002/jgt.20113 Abstract: A k-decomposition (G1, ...,Gk ) of a graph G is a partition of its edge set to form k spanning subgraphs G1, ...,Gk . The classical theorem of Nordhaus and Gaddum bounds (G1) þ (G2) and (G1)(G2) over all 2- decompositionsP of Kn. For a graph parameter p, let p(k;G) denote the k maximum of i¼1 p(Gi ) over all k-decompositions of the graph G. The clique number !, chromatic number , list chromatic number ‘, and Szekeres–Wilf number satisfy !(2;Kn) ¼ (2;Kn) ¼ ‘(2;Kn) ¼ (2;Kn) ¼ n þ 1. We obtain lower and upper bounds for !(k;Kn),(k;Kn),‘(k;Kn), and (k;Kn). The last three behave differently for large k. We also obtain lower and upper bounds for the maximum of (k;G) over all graphs embedded on a given surface. ß 2005 Wiley Periodicals, Inc. J Graph Theory 50: 273–292, 2005 Keywords: graph decomposition; Nordhaus–Gaddum Theorem; chromatic number; coloring number; list chromatic number; surface embedding 1. INTRODUCTION A k-decomposition of a (hyper)graph G is a decomposition of G into k spanning sub(hyper)graphs G1; ...; Gk. That is, each Gi has the same vertices as G, and every edge of G belongs to exactly one of G1; ...; Gk. Such decompositions can be viewed as unrestricted k-edge-colorings of G (color classes may be empty). For a parameter p, a positive integer k, and a (hyper)graph G, let noP k pðk; GÞ¼max i¼1 pðGiÞ: ðG1; ...; GkÞ is a k-decomposition of G : A p-optimalP k-decomposition of G is a k-decomposition ðG1; ...; GkÞ such that k Ppðk; GÞ¼ i¼1 pðGiÞ: We will also comment (in SectionQ 2) on the minimum of k k i¼1 pðGkÞ and on the extreme values of the product i¼1 pðGkÞ. The parameters we study are the clique number !, the chromatic number , the list-chromatic number ‘, and the Szekeres–Wilf number ,where is defined by ðGÞ¼1 þ maxHGðHÞ. Every graph G satisfies !ðGÞðGÞ‘ðGÞ ðGÞ. Therefore, for every k and G, !ðk; GÞðk; GÞ‘ðk; GÞðk; GÞ: ð1Þ Clearly, ðG1; G2Þ is a 2-decomposition of the complete graph Kn if and only if G1 is the complement of G2 and has n vertices. Thus the Nordhaus–Gaddum Theorem can be stated as follows. Theorem 1 (Nordhaus–Gaddum [16]). Let n be a positive integer. If ðG1; G2Þ is a 2-decomposition of Kn, then the following statements hold: NORDHAUS–GADDUM-TYPE THEOREMS 275 pffiffiffi (a) d2 n eðG1ÞþðGÀ2ÞÁn þ 1; nþ1 2 (b) n ðG1ÞðG2Þb 2 c: The proof of Theorem 1 implies that !ð2; KnÞ¼ð2; KnÞ¼‘ð2; KnÞ¼ ð2; KnÞ¼n þ 1, achieving equalities in (1) (see also Section 5). Finck [9] des- cribed the -optimal 2-decompositions of Kn. ðÞkþ1 Plesnı´k [17] studied ðk; KnÞ for k > 2 and proved that ðk; KnÞn þ 2 2 for every n. He also proved (see Lemma 3 in [17]) that k k n þ !ðk; K Þðk; K Þ if n ð2Þ 2 n n 2 ÀÁ k and conjectured (see also Bosa´k [3]) that ðk; KnÞ¼n þ 2 . Our first result is the following. ÀÁ ÀÁ k k Theorem 2. If k andÀÁ n are positive integers, then !ðk; KnÞn þ 2 .Ifn 2 , k then !ðk; KnÞ¼n þ 2 . k! Watkinson [24] improved Plesnı´k’s upper bound to ðk; KnÞn þ 2.It follows that !ð3; KnÞ¼ð3; KnÞ¼n þ 3 for every n 3. Our second result improves Watkinson’s bound for large k. k Theorem 3. If k and n are positive integers, then ðk; KnÞn þ 7 . Thus for chromatic number, the ‘‘lower order term’’ is independent of n. This does not hold for list chromatic number. ÀÁ ‘þ1 Theorem 4. There exists a positive constant c such that, if k ¼ 2 and n ¼ ‘m with ‘ and m being integers greater than 1, then n ‘ðk; KnÞn þ ck ln pffiffiffi : k On the other hand, for all positive integers k and n, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ‘ðk; KnÞn þ 3k!ð1 þ 8n ‘nnÞ: Thus, the leading behavior of ‘ðk; KnÞ for fixed k as n !1is still n, but the additive term grows with n. The leading behavior for ðk; KnÞ is larger. Theorem 5. If k ¼ p2 þ p þ 1 for some prime power p, and n 0 mod k, then pffiffiffi ðk; KnÞð k À 1Þn þ k: On the other hand, for all positive integers k and n, pffiffiffi ðk; KnÞ kn þ k: 276 JOURNAL OF GRAPH THEORY Furthermore, we determine ðk; KnÞ exactly for k 4. r We consider also decompositions of the complete r-uniform hypergraph Kn. r r Since ðKnÞ¼dn=ðr À 1Þe,wehaveðk; KnÞdn=ðr À 1Þe þ k À 1. Although it is easy to improve the bound for large n, we will prove that up to a summand r ck;r independent of n, this is the correct value of ðk; KnÞ. Theorem 6. If k and r are positive integers with r 2, then there exists an integer ck;r such that, for every positive integer n, n ðk; KrÞ þ c : n r À 1 k;r Finally, we consider decompositions of graphs embedded on a given surface Æ. For a graph parameter p and positive integer k, let pðk;Þ denote the maximum of pðk; GÞ over all graphs G embeddable on Æ. Let g be the Euler genus of Æ.We show that for fixed k and large g, the values of !ðk;Þ, ðk;Þ, ‘ðk;Þ, and ðk;Þ are asymptotically equal, unlike !ðk; KnÞ and ðk; KnÞ. Q In the next section, we comment on the analogous problems for maximizing k i¼1 pðGiÞ and for minimizing the sum or product. Section 3 then introduces terminology we use in discussing the problems for maximum sum. In subsequent sections, we treat consecutively the clique number, the chromatic number, the list chromatic number, and the Szekeres–Wilf number of graphs. The last two sec- tions are devoted to the chromatic number of r-uniform hypergraphs and to de- compositions of graphs embedded on surfaces. 2. MAXIMUM PRODUCT AND MINIMUM SUM To study the problem of maximum product for a parameter p, define Q  k p ðk; GÞ¼maxf i¼1pðGiÞ: ðG1; ...; GkÞ is a k-decomposition of Gg: Using the Cauchy–Schwarz Inequality, our bounds on pðk; KnÞ yield correspond-  ing bounds on p ðk; KnÞ.Forfixedk,    k ! ðk; KnÞ ðk; KnÞ‘ ðk; KnÞð1 þ oð1ÞÞðn=kÞ ; pffiffiffi  k ðk; KnÞð1 þ oð1ÞÞðn= kÞ ; and  r k ðk; KnÞð1 þ oð1ÞÞðn=kðr À 1ÞÞ : r Decomposing Kn and Kn into almost disjoint complete subgraphs of about the same size (pairwise sharing at most r À 1 vertices) shows that except for  the bound on , these upper bounds are asymptoticallyp tightffiffi p forffiffiffi fixed k. The  À k k construction in Theorem 5 yields that ðk; KnÞe ðn= kÞ : Thus, for  ðk; KnÞ, we at least know the order of magnitude. NORDHAUS–GADDUM-TYPE THEOREMS 277 We can also study the minimum sum or product of the values of p over a decomposition. Let noP k pðk; GÞ¼min i¼1 pðGiÞ: ðG1; ...; GkÞ is a k-decomposition of G ; and noQ  k p ðk; GÞ¼min i¼1 pðGiÞ: ðG1; ...; GkÞ is a k-decomposition of G : Comparing the arithmetic mean and the geometric mean, we infer that pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k k pÂðk; GÞ pðk; GÞ for every integer k 2 and every hypergraph G. Now consider ðk; GÞ and Âðk; GÞ, the lower bounds for the sum and the product of the chromatic numbers in a k-decomposition. We apply the natural extension of the argument of Nordhaus andQ Gaddum [16]. If ðG1; ...; GkÞ is a k- k decomposition of a graph G, then ffiffiffiðGÞ i¼1 ðGiÞ.
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