Enropy compression mehod applied o graph colorings Daniel Gonçalves a, Mickaël Monassier b, and Aleandre Pinlou c a CNRS, LIRMM b Universié Monpellier 2, LIRMM c Universié Monpellier 3, LIRMM 6 rue Ada, 34095 Monpellier Cede 5, France {daniel.goncalves,mickael.monassier,aleandre.pinlou}@lirmm.fr June 7, 204 Absrac Based on he algorihmic proof of Lovász local lemma due o Moser and Tardos, Espere and Parreau developed a framework o prove upper bounds for several chromaic numbers in paricular acyclic chromaic inde, sar chromaic number and Thue chromaic number using he so-called enropy compression mehod. Inspired by his work, we propose a more general framework and a beer analysis. This leads o improved upper bounds on chromaic numbers and indices. In paricular, every graph wih maimum degree has an acyclic chromaic number a mos 3 2 4 3 + O, and a non-repeiive chromaic number a mos 2 +.89 3 5 4 + O 3. Also every planar graph wih maimum degree has a facial Thue chromaic number a mos + O 2 and a facial Thue chromaic inde a mos 0. Inroducion In he 70 s Lovász inroduced he celebraed Lovász Local Lemma LLL for shor o prove resuls on 3-chromaic hypergraphs [2]. I is a powerful probabilisic mehod o prove he eisence of combinaorial objecs saisfying a se of consrains. Since hen, his lemma has been used in many occasions. In paricular, i is a very efficien ool in graph coloring o provide upper bounds on several chromaic numbers [3, 5, 4, 7, 20, 2, 24, 25]. Recenly Moser and Tardos [26] designed an algorihmic version of LLL by means of he so-called Enropy Compression Mehod. This mehod seems o be applicable whenever LLL is, wih he benefis of providing igher bounds. For eample, he Enropy Compression Mehod has been used in graph coloring o ge bounds on non-repeiive coloring [0] ha improve previous resuls using LLL see e.g. [3] and on acyclic-edge coloring []. In his laer paper, Espere and Parreau provide a general mehod applicable o many graph colorings. Inspired by his work, we provide a more general mehod and give new ools o improve he analysis. The paper is organized as follows. In Secion 2, we presen he mehod and apply i o acyclic vere coloring. I will be he occasion of providing improved bounds in erms of he maimum degree. Then, in Secions 3 and 4, we describe a general mehod and provide is analysis. Finally, in Secion 5, we apply his mehod o coloring problems such as generalized acyclic coloring problem, non-repeiive coloring problem, 2, F-subgraph coloring problem,... This research is parially suppored by he ANR EGOS, under conrac ANR-2-JS02-002-0.
2 Acyclic coloring of graphs A proper coloring of a graph is an assignmen of colors o he verices of he graph such ha wo adjacen verices do no use he same color. A k-coloring of a graph G is a proper coloring of G using k colors ; a graph admiing a k-coloring is said o be k-colorable. An acyclic coloring of a graph G is a proper coloring of G such ha G conains no bicolored cycles ; in oher words, he graph induced by every wo color classes is a fores. Le χ a G, called he acyclic chromaic number, be he smalles ineger k such ha he graph G admis an acyclic k-coloring. Acyclic coloring was inroduced by Grünbaum [8]. In paricular, he proved ha if he maimum degree is a mos 3, hen χ a G 4. Several aricles sudied graphs wih small maimum degree [2, 8, 9, 3, 5, 22, 34, 35, 36] and he curren knowledge is ha graphs wih maimum degree 4, 5, and 6, respecively verify χ a G 5, 7, and [8, 22, 23]. For higher values of, Kosochka and Socker [23] showed ha χ a G + + 2 4. Finally, for large values of he maimum degree, Alon, McDiarmid, and Reed [4] used LLL o prove ha every graph wih maimum degree saisfies χ a G 50 4/3. Moreover hey proved ha here eis graphs wih maimum degree such ha heir acyclic chromaic number is a leas 4/3. Recenly, log 4/3 he upper bound was improved o 6.59 4 3 + 3.3 by Ndreca e al. [27] and hen o 2.835 4 3 + by Sereni and Volec [32]. We improve his upper bound for large by a consan facor. Theorem Every graph G wih maimum degree 24 is such ha { } 3 χ a G < min 2 4 3 3 + 5 4, 2 4 8 4 3 3 + + 2 3 4 +. A he end of Secion 2.2. see Remark 0, we give a mehod o refine hese upper bounds, improving on Kosochka and Socker s bound as soon as 27. Alon, McDiarmid, and Reed [4] also considered graphs having no copy of K 2,γ+ he complee biparie graph wih parie ses of size 2 and γ + in which he wo verices in he firs class are non-adjacen. Le K γ be he familly of such graphs. Again using LLL, hey proved ha every graph G K γ wih maimum degree saisfies χ a G 32 γ. Using similar echniques as Theorem, we obain: Theorem 2 Le γ be an ineger and G K γ wih maimum degree. We have χ a G < + + 2γ + 4. As i is simpler, le us sar wih he proof of Theorem 2. 2. Graphs wih resricions on K 2,γ+ s We prove Theorem 2 by conradicion. Suppose here eiss a graph G K γ wih maimum degree such ha χ a G + + 2γ + 4. Le κ be he unique ineger such ha + 2γ + 4 κ < + + 2γ + 4. We define an algorihm ha "ries" o acyclically color G wih κ colors. Define a oal order on he verices of G. 2.. The algorihm Le V = {, 2,..., κ} be a vecor of lengh, for some arbirarily large n = V G. The following algorihm akes he vecor V as inpu and reurns a parial acyclic coloring ϕ : V G {,, 2,..., κ} of G means ha he vere is uncolored and a e file R ha is called a record in he remaining of he paper. The acyclic coloring ϕ is necessarily parial since we ry o color G wih a number of colors less han is acyclic chromaic number. For a given vere v of G, we denoe by Nv he se of neighbors of v. 2
Algorihm : ACYCLICCOLORINGGAMMA_G Inpu : V vecor of lengh. Oupu: ϕ, R. for all v in V G do 2 ϕv 3 R newfile 4 for i o do 5 Le v be he smalles w.r. uncolored vere of G 6 ϕv V [i] 7 Wrie "Color \n" in R 8 if ϕv = ϕu for u Nv hen // Proper coloring issue 9 ϕv 0 Wrie "Uncolor, neighbor u \n" in R else if v belongs o a bicolored cycle of lengh 2k k 2, say v = u,..., u 2k hen // Bicolored cycle issue 2 for j o 2k 2 do 3 ϕu j 4 Wrie "Uncolor, cycle v = u,..., u 2k \n" in R 5 reurn ϕ, R Algorihm ACYCLICCOLORINGGAMMA_G runs as follows. Le ϕ i be he parial coloring of G afer i seps a he end of he i h loop. A Sep i, we firs consider ϕ i and we color he smalles uncolored vere v wih V [i] line 6 of Algorihm. We hen verify wheher one of he wo following evens happens: Even. G conains a monochromaic edge vu for some u line 8 of Algorihm ; Even 2. G conains a bicolored cycle of lengh 2k v = u, u 2,..., u 2k line of Algorihm. If such evens happen, hen we uncolor some verices including v in order ha none of he wo previous evens remains. Clearly, ϕ i is a parial acyclic coloring of G. Indeed, since Even is avoided, ϕ i is a proper coloring and since Even 2 is avoided, ϕ i is acyclic. Proof of Theorem 2. Le us firs noe ha he funcion defined by Algorihm ACYCLICCOLOR- INGGAMMA_G is injecive. This comes from he fac ha from each oupu of he algorihm, one can deermine he corresponding inpu by Lemma 3. Now we obain a conradicion by showing ha he number of possible oupus is sricly smaller han he number of possible inpus when is chosen large enough compared o n. The number of possible inpus is eacly κ while he number of possible oupus is oκ, as i is a mos + κ n oκ. Indeed, here are a mos + κ n possible +κ-colorings of G and here are a mos oκ possible records by Lemma 4. Therefore, assuming he eisence of a counereample G leads us o a conradicion. This concludes he proof of Theorem 2. 2..2 Algorihm analysis Recall ha ϕ i denoes he parial acyclic coloring obained afer i seps. Le us denoe by ϕ i V G he se of verices ha are colored in ϕ i. Le also v i, R i and V i respecively denoe he curren vere v of he i h sep, he record R afer i seps, and he inpu vecor V resriced o is i firs elemens. Observe ha as ϕ i is a parial acyclic κ-coloring of G, and as G is no acyclically κ-colorable, we have ha ϕ i V G, and hus v i+ is well defined. This also implies ha R has "Color" lines. Finally noe ha R i corresponds o he lines of R before he i + h "Color" line. 3
Lemma 3 One can recover V i from ϕ i, R i. Proof. By inducion on i. Trivially, V 0 which is empy can be recovered from ϕ 0, R 0. Consider now ϕ i, R i and le us ry o recover V i. I is hus sufficien o recover R i, ϕ i, and V [i]. As observed before, o recover R i from R i i is sufficien o consider he lines before he las i.e. he i h "Color" line. Then reading R i, one can easily recover ϕ i and deduce v i. Noe ha in he i h sep we wroe one or wo lines in he record: eacly one "Color" line followed by eiher nohing, or one "Uncolor, neighbor" line, or one "Uncolor, cycle" line. Indeed here canno be an "Uncolor, cycle" line following an "Uncolor, neighbor" line, as v would be uncolored by he algorihm before considering bicolored cycles passing hrough v. Le us consider hese hree cases separaely. If Sep i was a color sep alone, hen V [i] = ϕ i v i and ϕ i is obained from ϕ i by uncoloring v i. If he las line of R i is "Uncolor, neighbor u", hen V [i] = ϕ i u and ϕ i = ϕ i. If he las line of R i is "Uncolor, cycle u,..., u 2k ", hen V [i] = ϕ i u 2k and ϕ i is obained from ϕ i by coloring he verices u j for 2 j 2k 2 which were uncolored in ϕ i, in such a way ha ϕ i u j equals ϕ i u 2k if j mod 2, or equals ϕ i u 2k oherwise. Noe ha his is possible because in he i h loop, he algorihm uncolored neiher u 2k nor u 2k. This concludes he proof of he lemma. Le us now bound he number of possible records. Lemma 4 Algorihm ACYCLICCOLORINGGAMMA_G produces a mos oκ disinc records R. Proof. Since Algorihm ACYCLICCOLORINGGAMMA_G fails o color G, he record R has eacly "Color" lines. I conains also "Uncolor" lines of wo ypes: "neighbor" and "cycle". Le be he number of "Uncolor, neighbor" lines, and le k be he number of "Uncolor, cycle" lines, where he cycle has lengh 2k 2 k n/2. Observe now ha: For every "Uncolor, neighbor" sep, he algorihm uncolors previously colored vere ; for every "Uncolor, cycle" sep, where he cycle has lengh 2k, he algorihm uncolors 2k 2 previously colored verices. I follows ha: + 2 k n/2 2k 2 k Le us recall ha he mulinomial coefficien is defined for K = i l k i by: K K! = k, k 2,..., k l k!k 2!... k l! Le us coun he number #Seq, 2,... n/2 of possible sequences of "Color" "Uncolor, neighbor" "Uncolor, cycle" lines in he record, for fied, 2,..., n/2. By Equaion, le us define he non-negaive ineger 0 = k n/2 k. Since each "Uncolor" line follows a "Color" line, 0 is he number of "Color" lines no followed by an "Uncolor" line. As here are "Color" lines, here are 0 choices for seing he "Color" lines no followed by an "Uncolor" line. Then here are 0 choices for seing he "Color" lines followed by 4
an "Uncolor, neighbor" line. Following his reasoning, he number of possible sequences is given by: 0 #Seq, 2,..., n/2 0 0,, 2,..., n/2 0 2 0 i< n/2 i n/2 To compue he oal number of possible records, le us compue how many differen enries in he record a given "Uncolor" sep can produce. Observe ha: An "Uncolor, neighbor" line can produce differen enries in he record, according o he neighbor of v ha shares he same color. An "Uncolor, cycle" line involving a cycle of lengh 2k can produce as many differen enries in he record as he number of 2k-cycles going hrough v. Thus his number of enries is a mos 2 γ 2k 2 according o Claim 5. Claim 5 Lemma 3.2 of [4] Consider a graph G K γ wih maimum degree. For any vere u of G and any k 2, here are a mos 2 γ 2k 2 cycles of lengh 2k going hrough u. Consequenly, he number of differen records for fied, 0,,..., n 2 is bounded by he following funcion B : B 0,,..., n 2 = = 0,,..., n 2 C i 0,,..., n 2 i i n/2 2 k n/2 k 2 γ 2k 2 where C =, C i = 2 γ 2i 2 for 2 i n 2. Summing over all possible uples 0,,..., n 2 saisfying Equaion, he number of differen records #Rec is bounded by: #Rec 0,,..., n 2 B 0,,..., n 2 By Corollary 9 of Secion 4, we have ha for a sufficienly large, #Rec < + n 2 Q for Q = + i n2 C i si wih s = and s i = 2i 2 for 2 i n 2 he s i s saisfy Equaion 9 by Equaion and any real 0 <. We hus have: Q = + C + C i 2i 2 2 i n 2 Seing X = γ 2 +, we have: C X = γ 2 + C i X 2i 2 = γ 2 γ 2 + i γ QX < 2 + + γ 2 + + = + 2γ + 4 κ Finally, we have #Rec = oκ. This complees he proof of Lemma 4. 5
2.2 Graphs wih maimum degree We prove Theorem by conradicion. To do so, we prove ha χ a G < 3 2 4 3 +5 4 for 24 in Secion 2.2. and ha χ a G < 3 2 4 3 + + 8 4 3 2 3 4 + for 9 in Secion 2.2.2. Suppose here eiss a graph G wih maimum degree which is a counereample o Theorem. Define a oal order on he verices of G. Le Nu and N 2 u be respecively he se of neighbors and disance-wo verices of u. For each pair of non-adjacen verices u and v, le Nu, v = Nu Nv, and le degu, v = Nu, v. For each vere u of G, le he order u on N 2 u be such ha v u w if degu, v < degu, w, or if degu, v = degu, w bu v w. A couple of verices u, v wih v N 2 u is special if here are less han C s 4 3 C s is a consan o be se laer verices w such ha v u w. Tha is, u, v is special if and only if, v is in he C s 4/3 highes elemens of u. Noe ha he couple u, v may be special while he couple v, u may be non-special. Le us denoe Su N 2 u he se of verices v such ha u, v is special. By definiion, Su C s 4 3. 2.2. Firs upper bound By hypohesis, χ a G 3 2 4 3 + 5 4. Le κ be he unique ineger such ha 3 2 4 3 + 5 5 κ < 3 2 4 3 + 5 4. The algorihm Le V = {, 2,..., κ} be a vecor of lengh. Algorihm ACYCLICCOLORING_G akes he vecor V as inpu and reurns a parial acyclic coloring ϕ : V G {,, 2,..., κ} of G recall ha means ha he vere is uncolored and a record R. Algorihm ACYCLICCOLORING_G runs as follows. Le ϕ i be he parial coloring of G afer i seps a he end of he i h loop. A Sep i, we firs consider ϕ i and we color he smalles uncolored vere v wih V [i] line 6 of Algorihm 2. We hen verify wheher one of he four following evens happens: Even. G conains a monochromaic edge vu for some u line 8 of Algorihm 2 ; Even 2. G conains a special couple v, u wih u and v having he same color line of Algorihm 2 ; Even 3. G conains a bicolored cycle of lengh 4 v = u, u 2, u 3, u 4 line 4 of Algorihm 2 ; Even 4. G conains a bicolored pah of lengh 6 u, u 2 = v, u 3, u 4, u 5, u 6 wih u u 3 line 8 of Algorihm 2. If such evens happen, hen we modify he coloring i.e. we uncolor some verices as menioned in Algorihm 2 in order ha none of he four previous evens remains. Noe ha a some Sep i, for u and v wo verices of G such ha u, v is a special couple bu v, u is no, we may have ϕu = ϕv; his means ha u has been colored before v. Clearly, ϕ i is a parial acyclic coloring of G. Indeed, since Even is avoided, ϕ i is a proper coloring ; since Evens 3 and 4 are avoided, ϕ i is acyclic. Proof of Theorem. As in he proof of Theorem 2, we prove ha he funcion defined by ACYCLIC- COLORING_G is injecive see Lemma 6. A conradicion is hen obained by showing ha he number of possible oupus is sricly smaller han he number of possible inpus when is chosen large enough compared o n. The number of possible inpus is eacly κ while he number of possible oupus is oκ, as he number of possible + κ-colorings of G is + κ n and he number of possible records is oκ see Lemma 7. 6
Algorihm 2: ACYCLICCOLORING_G Inpu : V vecor of lengh. Oupu: ϕ, R. for all v in V G do 2 ϕv 3 R newfile 4 for i o do 5 Le v be he smalles w.r. uncolored vere of G 6 ϕv V [i] 7 Wrie "Color \n" in R 8 if ϕv = ϕu for u Nv hen // Proper coloring issue 9 ϕv 0 Wrie "Uncolor, neighbor u \n" in R else if ϕv = ϕu for u Sv hen // Special couple issue 2 ϕv 3 Wrie "Uncolor, special u \n" in R 4 else if v belongs o a bicolored cycle of lengh 4 v = u, u 2, u 3, u 4 hen // Bicolored cycle issue 5 ϕv 6 ϕu 2 7 Wrie "Uncolor, cycle u, u 2, u 3, u 4 \n" in R 8 else if v belongs o a bicolored pah of lengh 6 u, u 2 = v, u 3, u 4, u 5, u 6 wih u u 3 hen // Bicolored pah issue 9 ϕu 20 ϕv 2 ϕu 3 22 ϕu 4 23 Wrie "Uncolor, pah u, u 2, u 3, u 4, u 5, u 6 \n" in R 24 reurn ϕ, R 7
Algorihm analysis Recall ha ϕ i, v i, R i, and V i respecively denoe he parial acyclic coloring obained afer i seps, he curren vere v of he i h sep, he record R afer i seps, and he inpu vecor V resriced o is i firs elemens. We firs show ha he funcion defined by ACYCLICCOLORING_G is injecive. Lemma 6 V i can be recovered from ϕ i, R i. Proof. Firs noe ha a each sep of Algorihm 2, a line "Color" possibly followed by a line "Uncolor" is appended o R. We will say ha a sep which only appends a line "Color" is a color sep, and a sep which appends a line "Color" followed by a line "Uncolor" is an uncolor sep. Therefore, by looking a he las line of R, we know wheher he las sep was a color sep or an uncolor sep. We firs prove by inducion on i ha R i uniquely deermines he se of colored verices a Sep i i.e. ϕ i. Observe ha R necessarily conains only one line which is "Color"; hen v is he unique colored vere. Assume now ha i 2. By inducion hypohesis, R i obained from R i by removing he las line if Sep i was a color sep or by removing he wo las lines if Sep i was an uncolor sep uniquely deermines he se of colored verices a Sep i. Then a Sep i, he smalles uncolored vere of G is colored. If one of Evens o 4 happens, hen he las line of R i is an "Uncolor" line whose indicaes which verices are uncolored. Therefore, R i uniquely deermines he se of colored verices a Sep i. Le us now prove by inducion ha he pair ϕ i, R i permis o recover V i. A Sep, ϕ, R clearly permis o recover V : indeed, v is he unique colored vere and hus V [] = ϕ v. Assume now ha i 2. The record R i gives us he se of colored verices a Sep i, and hus we know wha is he smalles uncolored vere v a he beginning of Sep i. If Sep i was a color sep, hen ϕ i is obained from ϕ i in such a way ha ϕ i u = ϕ i u for all u v and ϕ i v =. By inducion hypohesis, ϕ i, R i permis o recover V i and V [i] = ϕ i v. If Sep i was an uncolor sep, hen he las line of R i allows us o deermine he se of uncolored verices a Sep i and herefore, we can deduce ϕ i. Then by inducion hypohesis, ϕ i, R i permis o recover V i. We obain V i by considering he following cases: If he las line is of he form "Uncolor, neighbor u", hen V [i] = ϕ i u. If he las line is of he form "Uncolor, special u", hen V [i] = ϕ i u. If he las line is of he form "Uncolor, cycle u, u 2, u 3, u 4 ", hen V [i] = ϕ i u 3. If he las line is of he form "Uncolor, pah u, u 2, u 3, u 4, u 5, u 6 ", hen V [i] = ϕ i u 4. Lemma 7 Algorihm ACYCLICCOLORING_G produces a mos oκ disinc records. Proof. As Algorihm ACYCLICCOLORING_G fails o color G, he record R has eacly "Color" seps. Furhermore, here are "Uncolor" seps of differen ypes. Le, 2, 3, 4 be he number of "Uncolor" seps of ype "neighbor", "special", "cycle", and "pah", respecively. Noe ha each "Uncolor" sep of ype "neighbor" resp. "special", "cycle", and "pah" uncolors resp., 2, 4 previously colored vere; hus + 2 + 2 3 + 4 4 corresponds o he number of uncolored verices during he eecuion of he algorihm and hen + 2 + 2 3 + 4 4. Moreover, a he end of he eecuion of he algorihm here are less han n colored verices, and hus + 2 + 2 3 + 4 4 < n. Therefore, we have: n < + 2 + 2 3 + 4 4 2 8
Le us coun he number #Seq, 2, 3, 4 of possible sequences of "Color", "Uncolor, neighbor", "Uncolor, special", "Uncolor, cycle", and "Uncolor, pah" seps in he record, for fied,..., 4. Le 0 = + 2 + 3 + 4. We have "Color" seps and during he eecuion of he algorihm, every "Uncolor" sep follows a "Color" sep. So he number of possible sequences is given by: 0 0 0 2 #Seq, 2, 3, 4 0 2 3 0,, 2, 3, 4 To compue he oal number of possible records, le us compue how many differen enries in he record a given "Uncolor" sep can produce. By considering vere v in ACYCLICCOLORING_G, observe ha: An "Uncolor" sep of ype "neighbor" can produce differen enries in he record, according o he neighbor of v ha shares he same color; an "Uncolor" sep of ype "special" can produce Sv C s 4 3 record, according o he vere u Sv ha shares he same color; differen enries in he an "Uncolor" sep of ype "cycle" can produce as many differen enries in he record as he number of 4-cycles going hrough v and avoiding Sv. We do no consider bicolored 4-cycles going hrough v and some vere u Sv, since we would have a "Uncolor, special u" sep insead. Thus his number of enries is bounded by 3 8 8C s according o he ne claim. Claim 8 Given a graph G wih maimum degree, for any vere v of G, here are a mos 8 3 8C s induced 4-cycles going hrough v and avoiding Sv. Proof. There are a mos 2 edges beween Nv and N 2 v. Le d be an ineger such ha degv, u d if and only if u Sv. Therefore, here are a leas d Sv edges beween Nv and Sv. Thus here are a mos 2 dc s 4 3 edges beween Nv and Sv = N 2 v \ Sv, and degv, u 2 dc s 4 3 3 u Sv One can see ha he se of induced 4-cycles passing hrough v and hrough some vere u N 2 v is in bijecion wih he pairs of edges {u, uy} wih y and {, y} Nv, u. Thus here are degv,u 2 such cycles. Summing over all verices in Sv, we can hus conclude ha his is less han he following value K = 2 u Sv degv, u2. As his funcion is quadraic in degv, u, and as here degv, u d, Equaion 3 implies ha K Kd for Kd = 2 2 dc s 4 3 d. By simple calculaion one can see ha he polynomial Kd is maimal for d = 3 2 2C s and we hus have ha K K 2 3 2C s = 8 3 8C s. This concludes he proof of he claim. a sep "Uncolor" of ype "pah" can produce as many differen enries in he record as he number of pahs P 6 = u, u 2, u 3, u 4, u 5, u 6 wih u 2 = v and u u 3. Thus his number of enries is bounded by 2 4 according o he ne claim. Claim 9 Given a graph G wih maimum degree, for any vere v of G, here are a mos 2 4 pahs u, u 2, u 3, u 4, u 5, u 6 of lengh 6 wih u 2 = v and u u 3. 9
Proof. Given vere v, here are 2 possibiliies o choose u and u 3, and hen candidaes for being vere u i+ once u i is known i 3. This clearly leads o he given upper bound. This implies ha for fied, 0,..., 4, he number of differen records is bounded by he following funcion B : B 0,..., 4 = C s 4 2 8 3 4 3 3 4 0,, 2, 3, 4 8C s 2 = 0,, 2, 3, 4 i 4 where C =, C 2 = C s 4 3, C 3 = 8 3 8C s, and C 4 = 2 4. Summing over all possible 5-uples 0,..., 4 saisfying Equaion 2, he number of differen records #Rec is bounded by: #Rec 0,..., 4 C i i B 0,..., 4. By Corollary 9 of Secion 4, we have ha, for p = 4 and large, #Rec < + 4 Q for Q = + i 4 C i si wih s = s 2 =, s 3 = 2 and s 4 = 4 he s i s saisfy Equaion 9 by Equaion 2 and any real 0 <. We hus have: Q = + C + C 2 + C 3 2 + C 4 4 Seing X = 2 2C s, we have ha: 3 4 Q = + C 2 + C 3 2 + C + C 4 3 C 2 X = 2C s 2Cs C 3 X 2 = C = C 4 X 3 = 8 4 2C 3 2 s 3 One can obain: QX = + C s 4 3 + 8C 3 2 s 2 + 32C 3 8C 3 2 2 s 2 s 2 + 2Cs In order o minimize 2Cs + C s, we se C s = 2 and we obain: 6 4 + 2 QX = 3 2 4 3 + 5 6 + 24 6 2 + 4 3 < 3 2 4 3 + 5 5 κ as soon as 24 5 Finally, for sufficienly large, we have #Rec = oκ. This complees he proof of Lemma 7. Remark 0 For small values of, noe ha seing C s = 2 is no opimal. Indeed he bes choice of C s is he value minimizing he righ erm of Equaion 4. For eample, for = 27, seing C s = 0.225 leads us o 94 colors insead of 242, already improving on Kosochka and Socker s bound + + 2 = 97. Acually one can observe in Table ha he opimal value of C s for a 4 given converges o 2 raher slowly. 4 0
Algorihm 3: ACYCLICCOLORING-V2_G Inpu : V vecor of lengh. Oupu: ϕ, R. for all v in V G do 2 ϕv 3 R newfile 4 for i o do 5 Le v be he smalles w.r. uncolored vere of G 6 ϕv V [i] 7 Wrie "Color \n" in R 8 if ϕv = ϕu for u Nv hen // Proper coloring issue 9 ϕv 0 Wrie "Uncolor, neighbor u \n" in R else if ϕv = ϕu for u Sv hen // Special couple issue 2 ϕv 3 Wrie "Uncolor, special u \n" in R 4 else if v belongs o a bicolored cycle of lengh 2k k 2, say u, u 2 = v, u 3,..., u 2k wih u u 3 hen // Bicolored cycle issue 5 for j o 2k 2 do 6 ϕu j 7 Wrie "Uncolor, cycle u,..., u 2k \n" in R 8 reurn ϕ, R
27 28 29 30 00 000 0000 00000 000000 C s 0.225 0.225 0.226 0.226 0.25 0.32 0.384 0.434 0.465 Table : Opimal values of C s for some given. 2.2.2 A beer upper bound for large value of Algorihm ACYCLICCOLORING-V2_G leads us o he following upper bound: χ a G < 3 2 4 3 + + 8 4 3 2 3 4 +. Le κ be he unique ineger such ha 3 2 4 3 + + 8 4 3 κ < 3 2 3 4 2 4 3 + + 8 4 3 + and le 2 3 4 C s = 2. We now briefly skech he proof. By considering v in Algorihm ACYCLICCOLORING- V2_G, observe ha: An "Uncolor" sep of ype "neighbor" can produce differen enries in he record. Se C =. An "Uncolor" sep of ype "special" can produce Sv 2 4 3 differen enries in he record, according o he vere u Sv ha shares he same color. Se C 2 = 2 4 3. Now consider cycles of lengh 2k, k 2. For cycles of lengh 4, here are a mos 4 8 3 induced 4-cycles going hrough v and avoiding Sv see Claim 8; we se C 3 = 4 8 3. Le k 3. Le us upper bound he number of 2k-cycles going hrough v ha may be bicolored. To do so, we coun he number of 2k-cycles u, u 2, u 3,..., u 2k wih u 2 = v, u u 3 such ha u, u 2k or u 2k, u is no special if boh u, u 2k and u 2k, u are special, hen u and u 2k canno receive he same color. There are a mos 2k 4 3 such cycles according o Claim. We se C 2k = 2k 4 3. Claim For k 3, here are a mos 2k 4 3 2k-cycles u, u 2, u 3,..., u 2k going hrough v wih v = u 2 and u u 3 such ha u, u 2k or u 2k, u is no special. Proof. As u u 3, given v, here are 2 possible u, u 3. Then knowing u i, here are a mos possible choices for u i+, 3 i 2k 2. Now le r, s be a non-special pair being eiher u, u 2k or u 2k, u. Hence s N 2 r\sr. Le d be he highes value of degr, u for u N 2 r\sr. Therefore, here are a leas d Sr edges beween Nr and Sr, and so a mos 2 d 2 4 3 edges beween Nr and N 2 r \ Sr. I follows ha d is a mos 2 2 3. Hence, here are a mos 2 2 3 possible choices for u 2k. This leads o he given upper bound. I remains o upper bound Q for some such ha 0 < : Q = + C + C 2 + C 3 2 + n/2 k 3 C 2k 2k 2 2 Q < 3 4 3 2 2k 3 4 3 + + 2k 4 2 3 4 3 k 3 2 Q < 3 4 3 2 4 8 4 3 3 + + κ as soon as 9 2 3 4 2
3 General mehod In he previous secion, we gave upper bounds on he acyclic chromaic number of some graph classes. To do so, we precisely analyzed he randomized procedure for a specific graph class and a specific graph coloring. The aim of his secion is o provide a general mehod ha can be applied o several graph classes and many graph colorings some applicaions of our general mehod are given in Secion 5. In he remaining of his secion, G is an arbirarily chosen graph. The aim of he general mehod is o prove he eisence of a paricular coloring of G using κ colors, for some κ. A parial coloring of G is a mapping ϕ : V G {,, 2,..., κ} means ha he vere is uncolored. Given a parial coloring ϕ, le ϕ denoes he se of verices colored in ϕ. 3. Descripion of Algorihm COLORING_G Given a vere v of G, le Fv denoe he se of forbidden parial colorings anchored a v. This se is such ha for any ϕ Fv he vere v is colored. Noe ha we can have Fu Fv see an eample below. A parial coloring ϕ of G is said o be allowed, if and only if,. eiher ϕ is empy none of he verices is colored, 2. or here eiss a colored vere v such ha ϕ / Fv and uncoloring v yields o an allowed coloring. In mos of he applicaions of he general mehod, we have ha Fu = Fv for any colored verices u and v, ha implies ha no allowed coloring ϕ belongs o a se Fv, for some vere v. However in some cases see discussion a he end of he subsecion here are allowed colorings ϕ such ha ϕ Fv, for some vere v. We aim now a proving he eisence of an allowed coloring of G using κ colors, for some κ see laer Equaion 7. We assume by conradicion ha G does no admi such an allowed coloring. In ha case, we will show ha Algorihm COLORING_G see Algorihm 4 defines an injecive mapping Corollary 6 from κ differen inpus for some o oκ differen oupus Lemma 7, a conradicion. Algorihm 4: COLORING_G Inpu : V = {, 2,..., κ} vecor of lengh. Oupu: ϕ, R. for all v in V G do 2 ϕv 3 R newfile 4 for i o do 5 v NeUncoloredElemenϕ 6 ϕv V [i] 7 Wrie "Color \n" in R 8 if ϕ Bv hen 9 j BadEvenTypev, ϕ 0 k BadEvenClass j v, ϕ for u UncolorSeBadEven j v, ϕ, k do 2 ϕu 3 Wrie "Uncolor, Bad Even j, k \n" in R 4 reurn ϕ, R 3
Algorihm COLORING_G consrucs a parial coloring ϕ of G. A crucial invarian of Algorihm COLORING_G is ha he parial coloring ϕ obained afer any ieraion of he main loop is allowed. A he beginning of each ieraion Algorihm COLORING_G chooses by NeUncoloredElemen an uncolored vere v. NeUncoloredElemenϕ: This funcion akes he se of colored verices of G in ϕ as inpu and oupus an uncolored vere unless all verices are colored. Then Algorihm COLORING_G colors v. This new coloring ϕ eiher verifies ϕ / Fv and consequenly ϕ is allowed, or ϕ Fv and in ha case ϕ is an almos allowed coloring since uncoloring v yields an allowed coloring. Hence, le us define hese forbidden colorings ha can be produced by Algorihm COLORING_G. A parial coloring ϕ of G is said o be a bad even anchored a v, if ϕ Fv and if he parial coloring ϕ, obained from ϕ by uncoloring v, is such ha - ϕ is an allowed coloring, - v is he vere oupu by NeUncoloredElemenϕ. We denoe Bv he se of bad evens anchored a v. I is clear ha Bv Fv. For mos of he applicaions one could avoid inroducing Bv and jus deal wih Fv, however his seems mandaory for he applicaion eposed in Secion 5.3. Afer coloring v in he main loop, if he curren coloring ϕ / Bv, hen COLORING_G proceeds o he ne ieraion. Observe ha in ha case ϕ remains allowed as epeced. Suppose now ha afer coloring v, he curren coloring ϕ Bv. Before going furher ino he descripion of COLORING_G, le us inroduce he following refinemens of he ses Bv. For some se T, each se Bv is pariioned ino T ses B j v where j T. We call he bad evens of B j v he ype j bad evens. We now refine again each se B j v. We pariion each B j v ino differen classes B k j v where k belongs o some se C jv of cardinaliy a mos C j, for some value C j depending only on ype j. The pariion ino classes mus be sufficienly refined in order o allow some properies of he funcion RecoverBadEven see below. Afer noicing ha he curren coloring ϕ belongs o Bv, COLORING_G deermines he values j and k such ha ϕ B k j v. Tha is done using he following wo funcions: BadEvenTypev, ϕ: When ϕ is a bad even of Bv, his funcion oupus he elemen j T such ha ϕ is a bad even belonging o B j v. BadEvenClass j v, ϕ for some j T : When ϕ is a bad even of B j v, his funcion oupus he elemen k C j v such ha ϕ is a bad even belonging o B k j v. Then COLORING_G uncolors he verices given by UncolorSeBadEven, and proceeds o he ne ieraion. A key propery of UncolorSeBadEven is o ensure ha he obained coloring i.e. obained afer uncoloring he verices given by UncolorSeBadEven is allowed as epeced. UncolorSeBadEven j v, ϕ, k for some j T : For any bad even ϕ of B k j v wih colored verices ϕ, his funcion oupus a subse S of ϕ of size s j for some value s j depending only on ype j, such ha uncoloring he verices of S in ϕ yields an allowed coloring. Ofen he propery of leading o an allowed coloring is easy o fulfill see Lemma 2. A se X of parial colorings of G is closed upward resp. closed downward if saring from any parial coloring of X, coloring resp. uncoloring any uncolored resp. colored vere leads o anoher coloring of X. Lemma 2 If every se Fu is closed upward, hen he se of allowed colorings is closed downward. Hence in ha case, for any ϕ Bv uncoloring a se S of verices, wih v S, leads o an allowed coloring. 4
Proof. Le us firs prove he firs saemen. Assume for conradicion ha here eiss an allowed coloring ϕ and a non-empy se S ϕ, such ha uncoloring he verices in S leads o a non-allowed coloring ϕ. As ϕ is allowed, here eiss an ordering v,..., v p, wih p = ϕ, of he verices in ϕ such ha he resricion of ϕ o verices v,..., v i, denoed ϕ i, does no belong o Fv i, for any i p. Le us denoe ϕ i he coloring obained from ϕ i by uncoloring he verices of S if colored. As ϕ is no allowed, here eiss a value j p such ha ϕ j Fv j. Bu as Fv j is closed upwards, his conradics he fac ha ϕ j / Fv j. Consider now he second saemen. For any ϕ Bv, uncoloring v leads o an allowed coloring by definiion of Bv. Then he proof follows from he fac ha allowed colorings are closed downward. Finally, o prove he injeciviy of COLORING_G, we need ha he following funcion eiss. RecoverBadEven j v, X, k, ϕ where X V G, k C j v, and ϕ is a parial coloring of G: The funcion oupus a bad even ϕ B k j v, such ha ϕ = X, and such ha 2 uncoloring UncolorSeBadEven j v, ϕ, k from ϕ one obains ϕ, if such parial coloring ϕ eiss. Moreover, he pariion ino classes of B j v mus be sufficienly refined so ha a mos one bad even ϕ fulfills hese condiions. Eample. Le us illusrae our general mehod wih he proofs of Secion 2 on acyclic verecoloring. In Subsecion 2., for Algorihm, he ses Fv are all he same. They conain every parial coloring of G wih a monochromaic edge or wih a bicolored cycle. Hence he colorings in Bv are he bad evens such ha v belongs o a monochromaic edge or o a properly bicolored cycle. Then one ype say E corresponds o monochromaic edges, and several ypes say C 2k correspond o bicolored cycles, one per possible lengh of he cycle. Then each ype is pariionned ino classes according o he acual monochromaic edge, or o he acual bicolored cycle, respecively. For he uncoloring process, one can noice ha he number of uncolored verices only depends on he ype of bad evens, s E = and s C 2k = 2k 2, and ha he se of uncolored verices only depend on he class i.e. he monochromaic edge or he bicolored cycle. Furhermore, as he ses Fv are closed upward and as he curren vere is always uncolored, a he end of each ieraion he parial colorings are always allowed by Lemma 2. Finally, as described in Subsecion 2. here eiss a funcion RecoverBadEven j for each ype of bad even j. In Subsecion 2.2, for Algorihms 2 and 3, he siuaion is no eacly he same. Here, he ses Fv are sill closed upward bu hey are no all he same. This is due o he bad even corresponding o he special couples. Indeed, if u, v is a special couple bu no v, u, hen he colorings where hese verices use he same color necessarily belong o Fu while some of hem do no belong o Fv. However, similarly o Algorihm, Algorihms 2 and 3 fi o he general framework we described above. 3.2 Algorihm COLORING_G and is analysis From he previous subsecion, we have ha for j T, C j and s j respecively denoe he number of ype j bad even classes, and he number of verices o be uncolored in case of a ype j bad even. We se Q = + C j sj 6 j T and le κ be he smalles ineger such ha κ > inf 0< Q, i.e. κ = + inf Q 0< In his subsecion, we prove he following: 7 5
Theorem 3 The graph G admis an allowed κ-coloring. From now on, we assume ha G does no admi an allowed κ-coloring, his will lead o a conradicion. Le V = {, 2,..., κ} be a vecor of lengh, for some arbirarily large. The algorihm COLORING_G see Algorihm 4 akes he vecor V as inpu and reurns a allowed parial coloring ϕ of G and a e file R called record. Le ϕ i, v i, R i, and V i respecively denoe he parial coloring obained by Algorihm COLORING_G afer i seps, he curren vere v of he i h sep, he record R afer i seps, and he inpu vecor V resriced o is i firs elemens. Noe ha he algorihm and he properies of UncolorSeBadEven j v, ϕ, k ensure ha each ϕ i is allowed. As ϕ i is an allowed parial κ-coloring of G and since G has no allowed κ-coloring by hypohesis, we have ha ϕ i V G and ha vere v i+ is well defined. This also implies ha R has "Color" lines. Finally noe ha R i corresponds o he lines of R before he i+ h "Color" line. Lemma 4 One can recover v i and ϕ i from R i. Proof. By inducion on i. Trivially, ϕ 0 = and v 0 does no eis. Consider now R i+ and le us show ha we can recover v i+ and ϕ i+. To recover R i from R i+ i is sufficien o consider he lines before he las i.e. he i + h "Color" line. By inducion hypohesis, one can recover ϕ i from R i. Observe ha v i+ = NeUncoloredElemenϕ i. Le X = ϕ i + v i+. If he las line of R i+ is a "Color" line, hen ϕ i+ = X. Oherwise, he las line of R i+ is an "Uncolor" line of he form "Uncolor, Bad Even j, k". Then, we have ϕ i+ = X \ UncolorSeBadEven j v i+, X, k. Tha complees he proof. Lemma 5 One can recover V i from ϕ i, R i. Proof. By inducion on i. Trivially, V 0 which is empy can be recovered from ϕ 0, R 0. Consider now ϕ i+, R i+ and le us ry o recover V i+. By inducion, i is hus sufficien o recover R i, ϕ i, and he value V [i + ]. As previously seen in he proof of Lemma 4, we can deduce R i from R i+. By Lemma 4, we know ϕ i and we have v i+ = NeUncoloredElemenϕ i. Noe ha in he i + h sep of COLORING_G, we wroe one or wo lines in he record: eacly one "Color" line followed eiher by nohing, or by one "Uncolor, Bad Even j, k" line. Le us consider hese wo cases separaely. If Sep i + was a color sep alone, hen V [i + ] = ϕ i+ v i+ and ϕ i is obained from ϕ i+ by uncoloring v i+. If he las line of R i+ is "Uncolor, Bad Even j, k", hen he funcion RecoverBadEven j v i+, ϕ i, k, ϕ i+ oupus he bad even ϕ i ha occured during his sep of he algorihm. Then we have ha V [i+] = ϕ i v i+ and ha ϕ i corresponds o he parial coloring obained from ϕ i by uncoloring v i+. This concludes he proof of he lemma. Corollary 6 The mapping V ϕ, R defined by Algorihm COLORING_G is injecive. Lemma 7 Algorihm COLORING_G produces a mos oκ disinc records R. Proof. Consider any eecuion of Algorihm COLORING_G. Since i fails o color G by hypohesis, G does no admi an allowed κ-coloring, is record R has eacly "Color" lines. I conains also "Uncolor" lines of differen ypes: le j, for any j T, be he number of "Uncolor, Bad Even j" lines. As for each "Uncolor, Bad Even j" sep he algorihm uncolors s j previously colored verices, we have ha: 6
s j j 8 j T Le us coun he number of possible sequences of "Color" "Uncolor, Bad Even "... "Uncolor, Bad Even p" lines in he record, for fied numbers j, wih j T. By Equaion 8, le us define he non negaive ineger 0 = j T j. Since each "Uncolor" line follows a "Color" line, 0 is he number of "Color" lines no followed by an "Uncolor" line. As here are "Color" lines, here are 0 choices for seing he "Color" lines no followed by an "Uncolor" line. Then here are 0 j choices for seing he "Color" lines followed by an "Uncolor, Bad Even j" line. Following his reasoning he number of possible sequences is a mos he mulinomial 0,..., j,.... Then le us noe ha any "Uncolor, Bad Even j" line can be compleed in a mos C j differen ways. Consequenly, he number of differen records for fied, 0, j, for j T, is bounded by he following funcion B : B 0,..., j,... = 0,..., j,... Summing over all possible uples 0,..., j,... saisfying Equaion 8, he number of differen records #Rec is bounded by: #Rec B 0,..., j,... 0,..., j,... By Corollary 9 of Secion 4, we have ha for a sufficienly large, #Rec < + T inf = oκ This complees he proof of he lemma. 0< Q Proof of Theorem 3. Firs observe ha Algorihm COLORING_G can produce a mos oκ disinc oupus ϕ, R; indeed, here are a mos + κ n parial coloring ϕ of G and a mos oκ records R by Lemma 7. This is less han he κ possible inpus for a sufficienly large, and hus conradics he injeciviy of Algorihm COLORING_G Corollary 6. This concludes he proof. j T C j j 3.3 Eension o lis-coloring Given a graph G and a lis assignmen Lv of colors for every vere v of G, we say ha G admis a L-coloring if here is a vere-coloring such ha every vere v receives is color from is own lis Lv. A graph is k-choosable if i is L-colorable for any lis assignmen L such ha Lv k for every v. The minimum ineger k such ha G is k-choosable is called he choice number of G. The usual coloring is a paricular case of L-coloring all he liss are equal and hus he choice number upper bounds he chromaic number. This noion naurally eends o edge-coloring and chromaic inde. Unil now, our mehods were developed for usual colorings i.e. wihou liss. Every algorihm akes a vecor of colors V as inpu and, a each Sep i, a vere v is colored wih color V [i] line 6 of COLORING_G. I is easy o slighly modify our procedure o eend all our resuls o lis-coloring. To do so, he inpu vecor V is no longer a vecor of colors bu a vecor of indices. Then, a each Sep i, he curren vere v is colored wih he V [i] h color of Lv. We hen adap he proof of Lemma 5 so ha V [i + ] is no longer ϕ i+ v i+ or ϕ i v i+ bu insead i is he posiion of ϕ i+ v i+ or ϕ i v i+ in Lv i+. Therefore, Theorems, 2, and 3 eend o lis-coloring. 7
4 An upper bound for he funcion B In Secion 2, we inroduce he funcion B o coun he number of differen records ha Algorihm ACYCLICCOLORING_G may produce. In his secion, we generalize he definiion of B and we compue an upper bound. Le and p be wo posiive inegers. For i p, le s i be posiive inegers and C i be reals wih C i >. We consider a p + -ary funcion B of he form: B 0,..., p = C i i 0,,..., p 0 i p i p i p defined for non-negaive inegers 0,..., p such ha i = and s i i. 9 Noe ha we hen have 0 = i p i. Le Q : ]0, ] R be he funcion defined by Q = + C i si. i p Theorem 8 For sufficienly large, he maimum value of B is less han inf Q. 0< Moreover, we have: If s i = for all i p, hen inf Q = + C i. 0< i p Oherwise, inf Q = QX, where X is he unique posiive roo of he polynomial 0< P = + s i C i si. i p Roo X of Theorem 8 may be hard o compue. In such case, since Q QX for all 0 <, one can consider he upper bound of QX given by Q, for some 0 <. From Theorem 8, we ge he following corollary. Corollary 9 Summing over all possible p+-uples 0,..., p saisfying Equaion 9, we have for sufficienly large ha B 0,..., p < + p inf 0,..., p 0< Q Proof of Theorem 8. Le T 0,..., T p be he p + -uple maimizing B. Recall ha T 0 = i p T i. Le s = ma i p s i. Claim 20 If is sufficienly large, hen T i > 0 for every 0 i p. Proof. Le j be such ha T j is maimum among T 0,..., T p, and noe ha T j p+. We have T 0 > s. If j = 0, hen T 0 p+ > s since is chosen sufficienly large. Consider now he case j 0. We have T 0 + 0, T j p p+ 0 for sufficienly large, and by Equaion 9, i p s it i > s j + i p s it i. Then, he mapping B is also defined a T 0 +,..., T j,.... By definiion of T 0,..., T p, we hus have: B T 0,..., T j,... B T 0 +,..., T j,... 8
This is equivalen o: T 0! T j! C j T 0 +! T j! This implies ha T 0 Tj C j p+c j > s for a sufficienly large. 2 We have T k 3ps for some k wih k p. Assume T j < 3ps for every j wih j p. I follows ha i p T i i p s it i < 3. This also implies ha T 0 = i p T i > 3 = 2 3. Observe now ha, since T 0 0, T + 0, and s + i p s it i < s + 3, for sufficienly large, B is defined a T 0, T +,..., and we hus have, This is equivalen o: B T 0, T,... B T 0, T +,... T 0!T! T 0!T +! C This implies ha T T 0 C > 2 3 3 T k 3ps for some k wih k p. 3ps, a conradicion. Hence, we have Therefore, we have ha T 0 s + by and wihou loss of generaliy T 3 We have T k > 0 for every 2 k p. 3ps by 2. For all 2 k p, we have T 0 +s k s s++s k s +s k 0, T s k 3ps s k 0 when is sufficienly large, T k + s 0, and s T + s k T k = s T s k + s k T k + s Equaion 9 is saisfied, B is defined a T 0 + s k s, T s k,..., T k + s,..., and we hus have, B T 0, T,..., T k,... B T 0 + s k s, T s k,..., T k + s,... This is equivalen o: implying C s k T 0! T! T k! T k + s! T k! C s k T 0 + s k s! T s k! T k + s! Cs k C s k T 0! T! T 0 + s k s! T s k! T k + s s Cs k C s k Cs k C s k Observe now ha for large value of, we have T s k 3ps 3ps. I follows: T s k + s k T 0 + s k s s k s T s k s k T 0 + s k s k 4ps, and T 0 +s k as T 0 T T k C s k C s k 4ps s k s s = O s As he righ side of his inequaliy is sricly increasing wih, we hus have ha T k > 0 for sufficienly large. This concludes he proof of Claim 20. 9
Sirling s approimaion formula of k! [30] says ha for any k > 0: 2πk k+/2 e k < k! < 2πk k+/2 e k+/2k As T 0,..., T p, are posiive inegers by Claim 20, his implies he following bound. T 0,..., T p =! T 0! T p! < e/2 2π p/2 < e /2 2π p/2 +/2 T T0+/2 0 Tp Tp+/2 +/2 T T0 0 Tp Tp We hus have for sufficienly large ha B T 0,..., T p < + C 0 =. Le us define he p + -ary mapping ˆB : ˆB 0,..., p = + 0 i p 0 i p i p i Ci i 0 i p Ci T i Ti where we se defined for posiive reals 0,..., p such ha i = and s i i 0 i C Since lim i i 0 i =, we coninuously eend he definiion of ˆB o non-negaive reals. Therefore, ˆB is defined on he compac of [0, ] p+ fulfilling 0, and hus le X 0,..., X p be he p + -uple maimizing ˆB. Claim 2 For sufficienly large, X 0 > 0. Proof. Assume for conradicion ha X 0 = 0. Since = 0 j p X j, le i be he ineger verifying X i p > 0. If i = 0, hen he claim is rue. Assume now ha i > 0. For any sufficienly small real ε > 0, say ha ε <, we have ha X i ε > 0. Moreover, since X 0 + X i = X 0 + ε + X i ε, he mapping ˆB is also defined a X 0 + ε,..., X i ε,.... By definiion of X 0,..., X p we hus have: ˆB 0,..., X i,... ˆB ε,..., X i ε,... I follows: Ci Xi ε ε Ci Xi ε X i X i ε Xi Ci ε ε Ci X i ε X i ε ε ε ε Xi ε ε C i Xi ε X i C i + for sufficienly large. pc i + This conradics he fac ha ε <, and hus concludes he proof of he claim. 20
Claim 22 For sufficienly large, here eiss a non-negaive consan X such ha X = for every i p. Proof. This comes from he fac ha for any couple i, j [, p] 2 /si X i C ix 0 Xj /sj Xi /si C j X 0 C i X 0 We prove now Equaion. Consider in he following any couple i, j [, p] 2. If, for his given i, X i = 0, hen Eq. is saisfied. So suppose, for his given i, X i > 0. For any sufficienly small ε 0, we have: X 0 + εs j s i 0 as X 0 > 0 by Claim 2; X i εs j 0; s i X i + s j X j = s i X i εs j + s j X j + εs i ; X 0 + X i + X j = X 0 + εs j s i + X i εs j + X j + εs i. The four previous equaions imply ha ˆB is also defined a X 0 +εs j s i,..., X i εs j,..., X j + εs i,.... We hus have: ˆB X 0,..., X i,..., X j,... ˆB X 0 + εs j s i,..., X i εs j,..., X j + εs i,... I follows: X0 Xi Xj Ci Cj X 0 X i X j X0+εs j s i Ci Xi εs j Cj X 0 + εs j s i X i εs j X j + εs i Xj+εs i X 0 + εs j s i X0+εsj si X X0 0 X 0 + εs j s i X0/ε+sj si X X0/ε 0 X i εs j Xi εsj X Xi i X i εs j Xi/ε sj X Xi/ε i X j + εs i Xj+εsi X Xj j X j + εs i Xj/ε+si X Xj/ε j α/ε Since lim α+εβ ε 0 α = e β si Xj, he lef hand of his inequaliy ends o X 0 sj X 0 X i Cεsi j C εsj i Csi j C sj i and so Xj /sj Xi /si C j X 0 C i X 0 Noe ha since C i and X 0 are posiive by Claim 2, hese quoiens are defined. We can hus X /si, define he real value X by seing X = i C ix 0 for any i p. Finally noe ha since he X i are non-negaive, X is non-negaive. Claim 23 If s i = for every i p, hen X =. Proof. Noe ha, since we assume s i = for every i p, ˆB is defined for all non-negaive X 0,..., X p such ha = 0 i p X i. We firs prove ha X i X 0 for every i p. Since ˆB is defined a X 0,..., X i,..., ˆB is also defined a X i,..., X 0,.... Hence we have: ˆB X 0,..., X i,... ˆB X i,..., X 0,... 2
I follows: X0 Xi Ci Xi Ci X0 X 0 X i X i C Xi i C X0 i Since C i >, we have X i X 0. By Claim 2, we hus have X i > 0 for all 0 i p and we can consider he following wo inequaliies for sufficienly small ε: X 0 So, from he firs inequaliy, we obain: ˆB X 0,..., X i,... ˆB X 0 + ε,..., X i ε,... ˆB X 0,..., X i,... ˆB X 0 ε,..., X i + ε,... X0 Xi Ci X 0 X i X0+ε Ci Xi ε X 0 + ε X i ε X0 + ε X 0 X 0 ε X 0 + ε X i ε C i Xi ε X i X i ε The lef hand of he previous inequaliy ends o X0Ci X i when ε ends o 0. Hence X = X0Ci X i. From he second inequaliy, we have: X0 Xi Ci X 0 X i X0 ε Ci Xi+ε X 0 ε X i + ε X0 ε X 0 X 0 ε Xi + ɛ X 0 ɛ C i Xi + ε X i X i ε The lef hand of he previous inequaliy ends o Xi X 0C i when ε ends o 0. Hence, X = Xi X 0C i. Finally, from he wo inequaliies, we derive X = as claimed. Claim 24 If s k 2 for some k p, hen he bound given by Equaion 0 is igh, ha is = i p s ix i. Moreover in his case, X is he unique posiive roo of he polynomial P = + i p s i C i si, and X <. Proof. Wihou loss of generaliy assume s 2. For conradicion, assume > i p s ix i. Noe ha since = 0 i p X i we have X 0 > i p s i X i X. For any sufficienly small ε, X 0 ε 0, X +ε 0, and εs + i p s ix i. Hence, ˆB is defined a X 0 ε, X +ε,... and we have: I follows: ˆB X 0, X,... ˆB X 0 ε, X + ε,... X 0 X0 C X X X 0 ε X0 ε X + ε X+ε X X0 0 X0 X X0 ε X + ε X 0 X X X X + ε X 0 ε C ε X0 ε X+ε C X 0 ε X + ε ε > as C >. 22
Furhermore since X 0 > X, for sufficienly small ε X0 X X0 ε X + ε > X 0 X 0 X X0 ε X + ε X 0 ln + X ln X > 0 ε ε X 0 + X > 0 X 0 X A conradicion proving ha Equaion 0 is igh. Le us now prove he second par of he claim. Since we have X i = C i X 0 X si for every i p by Claim 22, one can derive ha X is a roo of P : X i = = s i X i 0 i p i p X 0 = s i X i i p X 0 = s i C i X 0 X si i p = s i C i X si i p Since s i 0 for all i p and s C >, he polynomial P is sricly increasing for 0, and so roo X is unique. Finally noe ha as s C >, we have X <. Since by Claim 22 we have Xi C i = X 0 X si for any i p, we can rewrie ˆB X 0,..., X p as follows: ˆB X 0,..., X p = + X X0 0 i p = X 0 X X 0 X si Xi where = i p s ix i. As and X by Claims 23 and 24, we can bound ˆB X 0,..., X p as follows: ˆB X 0,..., X p X 0 X Since = 0 i p X i = X 0 + i p C ix 0 X si, we have X 0 = + i p C ix si, and hus: ˆB X 0,..., X p + C i X si X To conclude we consider wo cases: QX i p Consider for all i p, s i =. Thus X = by Claim 23. In ha case noe ha Q is decreasing. Hence we derive: B X 0,..., X p < ˆB X 0,..., X p Q = inf Q 0< 23
Consider for some k p, s k 2. Noe ha P is an increasing funcion on [0, ] wih P 0 = and P > 0 recall ha s k 2 and C k >. Thus as Q = 2 P, i follows ha B X 0,..., X p < ˆB X 0,..., X p QX = inf Q 0< This concludes he proof of Theorem 8. 5 Some applicaions of he mehod o graph coloring problems In his secion, we apply he framework described in Secion 3 o differen coloring problems. We improve several known upper bounds by a leas an addiive consan and someimes also by a consan facor. More imporanly, his framework allows simpler proofs wih only few calculaions. Indeed, direcly using Theorem 3, one avoids he calculaions made in Secion 4. 5. Non-repeiive coloring In a vere resp. edge colored graph, a 2j-repeiion is a pah on 2j verices resp. edges such ha he sequence of colors of he firs half is he same as he sequence of colors of he second half. A coloring wih no 2j-repeiion, for any j, is called non-repeiive. Le πg be he non-repeiive chromaic number of G, ha is he minimum number of colors needed for any non-repeiive verecoloring of G. By eension, le π l G be he non-repeiive choice number of G. These noions were inroduced by Alon e al. [3] inspired by he works on words of Thue [33]. See [9] for a survey on hese parameers. Dujmović e al. [0] proved ha every graph G wih maimum degree saisfies π l G + + 2 = 2 + 2 5 3 3 + O 4 3 colors. Here we slighly 3 improve his formula, bu more imporanly, we provide a simple and shor proof. Theorem 25 Le G be a graph wih maimum degree 3. We have: π l G 2 + 3 5 2 2 3 5 3 2 2 3 + = 2 + 3 5 4 3 3 2 3 2 2 3 + O 3 3 3 2 2 3.89. Proof. To do his, le us use he framework as follows. Le G be any graph wih maimum degree, and le n denoe is number of verices. As in his applicaion, he ses Fv are closed upward we direcly proceed o he descripion of he bad evens as Fv is deduced from Bv, and he descripion of he required funcions. Le be any oal order on he verices of G. NeUncoloredElemenϕ reurns he firs uncolored vere according o. Le Bv be he se of bad evens ϕ anchored a v such ha vere v belongs o a repeiion in ϕ. The se Bv is pariioned ino subses B j v, for j n/2, in such a way ha in every ϕ B j v he vere v belongs o a 2j-repeiion. Le C j v be he se of 2j-vere pahs going hrough v. Each se B j v is pariioned ino subses B P j v according o he pah P C j v supporing he repeiion. If in a bad even ϕ Bv he vere v belongs o several repeiions, one of he repeiions is chosen arbirarily o se he value j and he pah P such ha ϕ B P j v. Le C j = j 2j as his upper bounds C j v. Indeed, here are 2j possible pahs on 2j verices where v has a given posiion, and 2j possible posiions for v, bu in ha case every pah is couned wice. 24