Linear choosability of graphs
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1 Liner hoosility of grphs Louis Esperet, Mikel Montssier, André Rspud To ite this version: Louis Esperet, Mikel Montssier, André Rspud. Liner hoosility of grphs. Stefn Felsner Europen Conferene on Comintoris, Grph Theory nd Applitions (EuroCom 05), 2005, Berlin, Germny. Disrete Mthemtis nd Theoretil Computer Siene, DMTCS Proeedings vol. AE, Europen Conferene on Comintoris, Grph Theory nd Applitions (EuroCom 05), pp , 2005, DMTCS Proeedings. <hl > HAL Id: hl Sumitted on 14 Aug 2015 HAL is multi-disiplinry open ess rhive for the deposit nd dissemintion of sientifi reserh douments, whether they re pulished or not. The douments my ome from tehing nd reserh institutions in Frne or rod, or from puli or privte reserh enters. L rhive ouverte pluridisiplinire HAL, est destinée u dépôt et à l diffusion de douments sientifiques de niveu reherhe, puliés ou non, émnnt des étlissements d enseignement et de reherhe frnçis ou étrngers, des lortoires pulis ou privés.
2 EuroCom 2005 DMTCS pro. AE, 2005, Liner hoosility of grphs L. Esperet, M. Montssier nd A. Rspud LABRI, Université Bordeux 1, Domine Universitire, 351 ours de l Liértion, Tlene, Frne A proper vertex oloring of non oriented grph G = (V, E) is liner if the grph indued y the verties of two olor lsses is forest of pths. A grph G is L-list olorle if for given list ssignment L = {L(v) : v V }, there exists proper oloring of G suh tht (v) L(v) for ll v V. If G is L-list olorle for every list ssignment with L(v) k for ll v V, then G is sid k-hoosle. A grph is sid to e linery k-hoosle if the oloring otined is liner. In this pper, we investigte the liner hoosility of grphs for some fmilies of grphs: grphs with smll mximum degree, with given mximum verge degree, plnr grphs... Moreover, we prove tht determining whether iprtite suui plnr grph is linery 3-olorle is n NP-omplete prolem. Keywords: vertex-oloring, list, yli, 3-frugl, hoosility under onstrints. 1 Introdution Let G e grph. Let V (G) e its set of verties nd E(G) its set of edges. A proper vertex oloring of G is n ssignment of integers (or lels) to the verties of G suh tht (u) (v) if the verties u nd v re djent in G. A k-oloring is proper vertex oloring using k olors. A proper vertex oloring of grph is liner if the grph indued y the verties of two olor lsses is forest of pths. The liner hromti numer, Λ(G), of G is the smllest integer k suh tht G is linery k-olorle. Liner oloring ws introdued y Yuster in [Yus98]. Using tehnis sed on the Lovász Lol Lemm, he proved: Theorem 1 [Yus98] Let G e grph with mximum degree, then Λ(G) = O( 3 2 ). A liner oloring of grph G is equivlent to n yli 3-frugl oloring of G: n yli oloring is vertex oloring suh tht the grph indued y the verties of two olor lsses is forest nd n 3-frugl oloring is vertex oloring suh tht the grph indued y the verties of two olor lsses is n indued sugrph with mximum degree 2, i.e. the neighourhood of ny vertex ontins the sme olor t most two times. The lower ound Λ(G) /2 + 1 is onsequene of the 3-fruglity. This ound is rehed for simple lsses of grphs, suh s trees. A greedy oloring of tree using deep-first serh gives ( /2 + 1)-oloring. The liner hromti numer of the omplete iprtite grph is lso esy to ompute: if m n, Λ(K m,n ) = m/2 + n. A grph G is L-list olorle if for given list ssignment L = {L(v) : v V (G)} there is oloring of the verties suh tht (v) L(v) nd (v) (u) if u nd v re djent in G. If G is L-list olorle for every list ssignment with L(v) k for ll v V (G), then G is sid k-hoosle. In this pper we fous on liner hoosility of grphs. This is, for whih vlue k, ny list ssignement L, with L(v) k for ll v V (G) llows liner oloring of G. Let Λ l (G) e the smllest integer k suh tht G is linery k-hoosle. We prove: Disrete Mthemtis nd Theoretil Computer Siene (DMTCS), Nny, Frne
3 100 Louis Esperet, Mikël Montssier nd André Rspud Theorem 2 Let G e n outerplnr grph with mximum degree, then Λ l (G) Theorem 3 Let G e grph with mximum degree, then: 1. If 3, then Λ l (G) If 4, then Λ l (G) 9. Let G e grph, the mximum verge degree of G, denoted y Md(G) is: Md(G) = mx{2 E(H) / V (H), H G}. Notie tht the mximum verge degree of grph n e omputed in polynomil time y using the Mtroid Prtitioning Algorithm due to Edmonds [Edm65, SU97]. Theorem 4 Let G e grph with mximum degree : 1. If 3 nd Md(G) < 16 7, then Λl (G) = If Md(G) < 5 2, then Λl (G) If Md(G) < 8 3, then Λl (G) Sine every plnr grph G with girth g(g) verifies Md(G) < 2g(G) g(g) 2, for plnr grphs we otin: Corollry 1 Let G e plnr grph with mximum degree : 1. If 3 nd g(g) 16, then Λ l (G) = If g(g) 10, then Λ l (G) If g(g) 8, then Λ l (G) 2 Oserve tht yles re linery 3-hoosle; hene, we nnot remove the ondition on in Theorem 4.1. Theorem 5 Let G e plnr grph with mximum degree 9, then Λ l (G) Theorem 6 Deiding whether iprtite suui plnr grph is linery 3-olorle is n NP-omplete prolem. 2 Proof skethes In the following, k-vertex (resp. k-vertex, k-vertex) is vertex of degree k (resp. k, k). Sketh of proof of Theorem 2 In [BGH03], Bonihon, Gvoille nd Hnusse show tht n outerplnr grph n e deomposed into spnning tree nd set of edges M. Eh edge of M links vertex v with the vertex f(v) defined s follows : if v hs rother t its left, then f(v) is v s rightmost left rother. Else f(v) = f(u), where u is v s fther (see Figure 1.). We olor the verties of G greedily using deep-first serh in the spnning tree. At eh step of the lgorithm, the vertex tht we re trying to olor hs t most /2 + 1 olors foridden y its neighourhood (t distne t most 2). Thus, there is t lest one olor in the list of the urrent vertex tht is not foridden, nd this greedy lgorithm gives liner oloring of G given lists of size t lest / During the proess, we distinguish two types of verties : those whih do not hve ny rother t their left (Type 1 verties, tht n rete ihromti yles), nd the others (Type 2 verties).
4 Liner hoosility of grphs 101 u u w v v w ) ) ) Figure 1: ) An exmple of the deomposition of n outerplnr grph. ) A vertex of Type 1. ) A vertex of Type 2. Sketh of proof of Theorem 3.1 The proof is sed on the method of the reduile onfigurtions. Let G e ounterexmple with minimum order. We prove tht G does not ontin the onfigurtions depited in Figure 2, thus otining ontrdition. Figure 2: The reduile onfigurtions. The proof of Theorem 3.2 lso uses the method of reduile onfigurtions. Sketh of proof of Theorem 4.2 Let H e ounterexmple of minimum order with Md(G) < 5 2. By minimlity, H does not ontin the following onfigurtions: 1. 1-verties, 2. two djent 2-verties, 3. 3-vertex djent to three 2-verties. We omplete the proof of Theorem 4.1 with dishrging proedure. First, we ssign to eh vertex v hrge ω(v) equl to its degree. We then pply the following dishrging rule: Rule 1 Eh 3-vertex gives 1 4 to eh djent 2-vertex. Let ω (v) e the new hrge of the vertex v fter the dishrging proedure. Let v e k-vertex (with k 2, sine H does not ontin ny 1-verties): If k = 2, then ω(v) = 2 nd ω (v) = = 5 2 sine v is not djent to ny 2-vertex nd v reeives 1 4 from eh djent vertex y Rule 1.
5 102 Louis Esperet, Mikël Montssier nd André Rspud If k = 3, then ω(v) = 3 nd ω (v) sine v is djent to t most two 2-verties nd thus gives t most y Rule 1. If k 4, then ω(v) = k nd ω (v) k k 1 4 3k 4 > 5 2 sine v my e djent to k 2-verties. So, v V (H), ω (v) 5 2. Now, oserve tht 2 E(H) = v V (H) ω(v) = v V (H) ω (v). Then, y definition of the mximum verge degree, we hve : Md(H) 2 E(H) v V (K) = ω (v) 5/2 V (H) = 5 V (H) V (H) V (H) 2 The ontrdition with Md(H) ompletes the proof of Theorem 4.2. The proof of Theorems 4.1 nd 4.3 re sed on the sme method nd on similr rguments. Sketh of proof of Theorem 5 The proof of Theorem 5 is sed on vn den Heuvel nd MGuinness struturl lemm [vdhm03] nd on the method of reduile onfigurtions. Sketh of proof of Theorem 6 Let us first show the existene of speil iprtite suui plnr grph E(x, y) (see Figure 3). The grph E(x, y) is suui, plnr, nd iprtite. Moreover, ny liner 3-oloring of E(x, y) verifies: (x) = (y). x y Figure 3: The grph E(x, y) : ny liner 3-oloring of E(x, y) verifies: (x) = (y) The proof of the NP-ompleteness proeeds y redution to the prolem of 3-oloring of plnr grphs, tht is n NP-omplete prolem [GJS76]. Given n instne of this prolem plnr grph G, we need to rete iprtite suui plnr grph H of size polynomil in V (G) suh tht H is linery 3-olorle if nd only if G is 3-olorle. We onstrut H s follows: For ll k-vertex u of G, we reple u y inry tree T u with k leves in H suh tht eh edge xy of T u is repled y E(x, y). Now, for ll djent vertex w of u in G, we ssoite one lef u w in H. Finlly, for eh edge uv of G, we dd in H n edge linking u v nd v u (see Figure 4). Suppose now tht there exists 3-oloring of G. By pplying the olor of the vertex v to the root of the tree T v in H, it is esy to extend the oloring to otin liner 3-oloring of H (rell tht if the root of T v hs the olor, then every lef of T v hs the olor nd given the olor of the root of T, it exists liner 3-oloring of T ). Conversely, if it exists liner 3-oloring of H, it exists 3-oloring of G y pplying the olor of the root of the tree T v in H to the vertex v in G.
6 Liner hoosility of grphs 103 v u v v u u v u T v T u 3 Conlusion We onlude with some open prolems: Figure 4: The onstrution of H from G Prolem 1 Is it true tht every suui grph different from K 3,3 is linery 4-olorle? Prolem 2 Minimize the rel vlues 1 nd suh tht every plnr grph with mximum degree dmits liner ( + )-oloring? Referenes [BGH03] N. Bonihon, C. Gvoille, nd N. Hnusse. Cnonil deomposition of outerplnr mps nd pplition to enumertion, oding, nd genertion. In WG2003: 29 th Interntionl Workshop, Grph - Theoreti Conepts in Computer Siene, pges 81 92, [Edm65] [GJS76] [SU97] J.R. Edmonds. Minimum prtition of mtroid into independent susets. J.Res, Nt. Bur. Stndrds, 69B:67 72, M.R. Grey, D.S. Johnson, nd L.J. Stokmeyer. Some simplified NP-omplete grph prolems. Theor. Comput. Si., 1(3): , E.R. Sheiermnn nd D.H. Ullmn. Frtionl Grph Theory: A Rtionl Approh to the Theory of Grphs. Wiley-Intersiene series, [vdhm03] J. vn den Heuvel nd S. MGuinness. Coloring the squre of plnr grph. Journl of Grph Theory, (42): , [Yus98] R. Yuster. Liner oloring of grphs. Disrete Mth., (185): , 1998.
7 104 Louis Esperet, Mikël Montssier nd André Rspud
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