Square-free Walks on Labelled Graphs

Square-fr ee W alks on Labelled Graphs T ero Harju Department of Mathemati cs Univ ersity of T urku, F inland harju@utu. fi June 8, 2018 Abstract A finite or in finite word is called a G -w ord for a labelled graph G o n the vertex set A n = { 0 , 1 , . . . , n − 1 } if w = i 1 i 2 · · · i k ∈ A ∗ n , wh ere eac h factor i j i j +1 is an edge of E , i.e , w represents a walk in G . W e show that there exists a square- free infinite G -word if and on ly if G has n o subgrap h isomorph ic to one o f th e cycles C 3 , C 4 , C 5 , the path P 5 or the claw K 1 , 3 . The colo ur nu mber γ ( G ) o f a gr aph G = ( A n , E ) is the smallest integer k , if it exists, f or which there exists a m apping ϕ : A n → A k such that ϕ ( w ) is squar e-free for an in finite G -word w . W e show that γ ( G ) = 3 f or G = C 3 , C 5 , P 5 , but γ ( G ) = 4 for G = C 4 , K 1 , 3 . In par ticular , γ ( G ) ≤ 4 for all graphs that have at least fi ve v ertices. Note The auth or is t hankfu l for James Currie for p ointing out that, apart from some techni cal dif fere nces, the results of the present paper are covere d by his paper James D. Cu rrie: Which gr aphs allo w infinite non repeti ti ve walks? Discr ete M ath. 87 (1991) 249–26 0. Ke ywords: Square-fr ee words , infinite words, G -word s, graphs. 1 Introd uction Alon et al . [1] initiate d the study of non-re peti v eness of simple pa ths in edge colour ed graphs. In [1] an edge colouri ng of a graph G is square-free if the se- quenc e of colours on ev ery simpl e pa th in G cons ists of a squ are-free word . The proble m for vert ex colo ured graphs was consider , e.g., by Bar ´ at and V arj ´ u [2]. For results on these settings, see also [3, 6, 5, 9]. In the prese nt paper we consider the proble m when a v erte x colouring of a graph has an infinite square-free walk. W e consider squ are-fre eness of infini te w or ds w : N → A n ov er the alphabets A n = { 0 , 1 , . . . , n − 1 } of cardinali ty n . An infinite word w will be represen ted as a seq uence of lett ers from A n , i.e., w = i 1 i 2 · · · with i j ∈ A n for all j ∈ N . 1 A (fi nite) w or d is a finite sequ ence of letters. The set of all finite wor ds o ver A n is denoted by A ∗ n . This set contain s the empty word . The length of a word w is denote d by | w | . If w = u 1 v u 2 , then v is a factor of w . If here u 1 is the empty word, then v is a pr efix of w and if u 2 is empty , then v is a suffix of w . The abov e terminol ogy generalizes to infinite words in a na tural way . A finite o r infinite word w is squ ar e- fr ee if it do es not conta in any f actor of th e form u 2 = uu for a no nempty word u . Thue [1 2] sho wed a hun dred years ago th at there are infini te squar e-free words ov er th e ternary alpha bet A 3 ; see Lot haire [11] or the historic al surv ey by Berstel and Perrin [4] . An ex ample o f an infinite square- free word can be obt ained by iterating the morphism τ (0) = 012 , τ (1) = 02 , τ (2) = 1 startin g from the letter 0 ∈ A 3 . The resu lt is the infinite word t = 01202 10121 02012021020121012 · · · . (1) Observ e that t does not hav e facto rs 010 and 212 . W e call t the Thue wor d al- thoug h it is due to Istra il [10]. A mapping α : A ∗ → B ∗ between word set s o ve r the alp habets A and B is a morphism , if α ( uv ) = α ( u ) α ( v ) for all word s u, v ∈ A ∗ . Clearly , each morphi sm α : A ∗ → B ∗ is determin ed by its images α ( a ) of the letters a ∈ A . A morphism α is squar e-fr ee , if it preserve s square-fre eness of words, i.e., if v ∈ A ∗ is square- free, then so is the image α ( v ) ∈ A ∗ . Let G = ( A n , E ) be an undirected graph on the ver tex set A n such that the edge set E does no t conta in self-lo ops from a vert ex i to itself nor parallel edges between the same vertice s. A n edge between i and j is denoted by ij . W e say that a word w = i 1 i 2 · · · i k ∈ A ∗ n is a G - wor d if each fac tor i j i j +1 is an edge of E . Thus a G -word is a wa lk on G . A k - colouring of a gr aph G on A n is a morph ism ϕ : A ∗ n → A ∗ k . The graph G has an infinite k -colour ed squar e-fr ee walk , if there is an infinite G -word w such that ϕ ( w ) is square-fr ee for some k -col ouring ϕ . W e denote by γ ( G ) the smallest k , if it exists , for which G has an infinite k -colo ured square-free G -word. Let P n denote the (s imple) path on the vert ices A n with the n − 1 edges i ( i + 1) , for i = 0 , 1 , . . . , n − 2 . Similarly , C n denote s the cy cle on A n with the n edges i ( i + 1) , where the in dices are modulo n − 1 . Finally , let K 1 , 3 be the gr aph on A 4 , called the claw , with the edges 01 , 02 , 03 . The follo wing the orem is our main result . It will be prov en in the ne xt sec tion. In particu lar , we sho w that, for connected graphs, there exist s an infinite square- free G -wor d if the graph G is a cycl e C m , for m ≥ 3 , or a path P m , for m ≥ 5 , or it conta ins the cla w K 1 , 3 as an induce d subgra ph. Theor em 1. Ther e e xists an in finite squar e-fr ee G -wor d if and o nly if the graph G has a conne cted subgrap h isomorph ic to one of the gra phs C 3 , C 4 , C 5 , P 5 , K 1 , 3 . 2 Mor eo ver , if G has a subgr aph C m with m = 3 or P n with n ≥ 5 , then γ ( G ) = 3 , and, otherwise , if G has G = C 4 or K 1 , 3 then γ ( G ) = 4 . Cor ollary 1. F or each graph G of at least five verti ces, γ ( G ) ≤ 4 . Pr o of. If G does not hav e a sub graph K 1 , 3 , then th e maxi mum deg ree of G is at most two, and in this case the connected compone nts of G are either paths P m or cyc les C k . The claim follo ws, since each path or cycle of five ve rtices has a subgra ph P 5 , C 3 , C 4 or C 5 . (Note that C k for k ≥ 6 contains P 5 .) 2 Pr oof of the theore m Recall that in genera l a subgra ph H of a graph G need not be induced, i.e., H might miss some edges of G that are between vert ices of H . Lemma 2. Let G be a gr aph. The followin g cases are e quival ent. (i) Ther e e xists an infini te squar e-f r ee G -wor d. (ii) G has a connecte d su bgr aph H that has an infin ite squar e-fr ee H -word . Pr oof. First, for eac h subgraph H of G , each squar e-free H -wo rd is also a square- free G -word. Secondly , if the ve rtices i and j of G belo ng to differe nt connected compone nts, then the letters i, j ∈ A n (where n is the order of G ) ne ver occur in the same G -wo rd. 2.1 Negative cases Recall tha t P 3 has onl y three vertices and two edges 01 and 1 2 . Hence eve ry second letter in a squ are-free P 3 -word w must be 1 , and hence the wo rd w ′ obtain ed by deletin g all occurre nces of 1 must be a binary squ are-fre e word, and thu s of lengt h at most three. The case for the graph P 4 is somewhat more complicat ed, bu t a systematic study , by hand or aided by a computer , quickly ends. The longest square-fre e P 4 - words are of length 15. They are 01210123 21012 10 and the dual 3212 32101 23212 3 obtain ed by the permut ation (0 3)(1 2) . Lemma 3. Ther e ar e no squar e-fr ee P 4 -wor ds of length 16. 2.2 The cases P m f or m ≥ 5 Consider the morphism α : A ∗ 3 → A ∗ 3 defined by α (0) = 201021 202101201021012021 of lengt h 24 , α (1) = 201021 2021012021 of lengt h 16 , α (2) = 201021 01 of lengt h 8 . 3 Note that the morphis m α is not squ are free, since α (010) = 201021 20210 12010210120(2120102120210120) 2 10210 12021 . Ho wev er , as we hav e seen, the Thue word t does not con tain 010 . The first case of the n ext l emma is seen to hold by checki ng the sho rt words by a computer program. T he second claim is obv ious, sin ce for i 6 = j , the word α ( i ) does not ha ve a suf fi x that is a prefix of α ( j ) , i.e., differ ent words α ( i ) and α ( j ) do not ov erlap. (Although α (2) is a factor of α (0) .) Lemma 4. (a) If w is of length 5 is squar e-fr ee and does not contain 010 , then also α ( w ) is squar e-fr ee. (b) The wor d α ( i ) with i ∈ { 0 , 1 } is aligned in α ( A 3 ) ∗ , i.e., if w = i 1 i 2 · · · i n ∈ A ∗ 3 and α ( w ) = u 1 α ( i ) u 2 then u 1 = α ( i 1 · · · i k − 1 ) , i = i k and u 2 = α ( i k +1 · · · i n ) . Lemma 5 . Let w be a squ ar e-fr ee wor d tha t doe s not h ave a fac tor 010 . Then also α ( w ) is a squar e fre e. F or the Thue wor d t , the infinit e wor d α ( t ) is a 3 -colour ed squar e-fr ee P 5 -wor d. Pr oof. Assume that α ( w ) co ntains a nonempty square, say , for so me u ∈ A ∗ 3 and letters p , p 1 , p 2 ∈ A 3 , α ( w ) = w 1 u 2 w 2 , w here u = v 1 α ( v ) v 2 = v ′ 1 α ( v ′ ) v ′ 2 , v 2 v ′ 1 = α ( p ) , v 1 is a suf fi x of α ( p 1 ) , v ′ 2 is a prefix of α ( p 2 ) . By Lemma 4(a), v and v ′ are nonempty , and by L emma 4(b), v = v ′ and also v 1 = v ′ 1 and v 2 = v ′ 2 . Now α ( p ) = v 2 v 1 , where v 1 is a suf fix and v 2 a su f fix of some word s α ( i ) and α ( j ) , respecti vely . One sees that this is not the case for any p, i, j , simply becaus e α ( p ) begins with the special word 2010 that occurs only as prefix of the words and onc e in the middle of α (0) . Therefore , α ( t ) is square-fr ee, si nce all its fi nite prefixe s are square- free and t does not ha ve any occur rences of the fa ctor 010 . For the second claim, consider the path P 5 with the edg es 01 , 12 , 23 , 34 , and let the colour ing of the vertices be ϕ : A 5 → A 3 defined by ϕ (0) = 1 , ϕ (1) = 0 , ϕ (2) = 2 , ϕ (3) = 0 , ϕ (4) = 1 . Then the morphism β : A ∗ 3 → A ∗ 5 defined by β (0) = 20102 320234320 1023432023 β (1) = 20102 320234320 23 β (2) = 20102 343 satisfies α = ϕβ . Since α ( t ) is square-fre e, so is β ( t ) . Clearly β ( t ) is a P 5 - word. Cor ollary 2. W e have γ ( G ) = 3 for all connected graph s co ntaini ng a subgraph P 5 . In p articul ar , for ea ch P n and C n with n ≥ 5 ther e e xists an infinite 3- colour ed squar e-fr ee P n -wor d. Pr oof. Indeed, each P n and C n , with n ≥ 5 , has a subgrap h equal to P 5 . Moreov er , we always hav e that if G cont ains a P 5 , then γ ( G ) ≥ 3 , since there are no infinite square -free binary words. Hence the first claim holds. 4 2.3 The case f or graphs with subgraphs C 3 , C 4 or K 1 , 3 W e first consi der the claw K 1 , 3 . Lemma 6. Let G be a graph with a verte x of de gr ee at least thr ee. Then ther e e xists an infinite 4 -colou r ed squar e-fr ee G -wor d. Mor eover , ther e does not exi st any infinit e 3 -colour ed squar e-fr ee K 1 , 3 -wor ds. Pr oof. W ithout loss of generalit y , we can assume that the verte x 3 of G has the neighb ours 0 , 1 , 2 . Let w be an infinite square-fr ee word w o ver A 3 , and consid er the infini te word w ′ obtain ed by replaci ng each i ∈ A 3 by the word i 3 of length two. It is clear th at w ′ is still square-fr ee and also that it is a G -word , where four colour s are prese nt. For the ne gati ve case, let w = ϕ ( u ) for a 3 -col ouring ϕ of G and a G -wor d u = 3 i 1 3 i 2 · · · . No w also ϕ ( i 1 i 2 · · · ) must be square-free . Ho wev er , we hav e two equic oloured neighbou rs i and j of the verte x 3 , say ϕ ( i ) = ϕ ( j ) , and thus ϕ ( i 1 i 2 · · · ) is a binary word, and not squa re-free at all. By Lemma 6 , we can assume that the maximum degre e of the graph G is two, and hence the connecte d co mponent s are either paths or cycles. By Lemma 3 and Corollary 2, this lea ves the connected graphs that are cycles C 3 or C 4 . Lemma 7. Ther e exis ts an infin ite squar e-fr ee C n -gr aph in all cases n ≥ 3 . W e have γ ( C 3 ) = 3 and γ ( C 4 ) = 4 . Pr oof. For n = 3 , we observe that eve ry square-fr ee word ov er the alphabet A 3 is also a C 3 -word . Thus the case for C 3 is clear . Let then n = 4 . If w ∈ A ∗ n − 1 is a squ are-free C n − 1 -word , then w ′ ∈ A ∗ n is a square-fr ee C n -word , where w ′ is obtained by inserting the letter n − 1 between e very pai r (0 , n − 2) and ( n − 2 , 0) . Ind eed, if w ′ has a squ are u 2 , th en th e del etion of n − 1 would yield a nonempty square in w ; a contrad iction . This sho ws that γ ( C 4 ) ≤ 4 . Consider then a 3-colourin g ϕ of C 4 , and assume ϕ ( w ) is a square -free C 4 - word. Then two diago nal ele ments of the square C 4 must obtai n the same colour , say ϕ (1 ) = ϕ (3) ; oth erwise the C 4 -word coloured by ϕ would be a P 4 -word . N o w , the colour ϕ (1) = ϕ (3) occurs in eve ry second place in ϕ ( w ) , and hence as in the proof of Lemma 6, the case reduc es to binary words, giv ing a contrad iction. W e ha ve now a simp lified proof of a result due to Dean [8]. Cor ollary 3 (Dean) . T her e exis ts an infinite squar e-fr ee wor d w that is r educed in the fr ee gr oup of two gener ators. Pr oof. W e apply Lemma 7 to C 4 . Let w be a square-f ree C 4 -word . One inter - prets 2 as the in ve rse element of th e generato r 0 and 3 as the in vers e element of the genera tor 1 . Since 0 and 2 , and 1 and 3 , respect i vel y , are nev er adja cent in w the word w is reduced in the free group. 5 W e also can sho w the exist ence of an infinite square- free C 4 -word by cons ider - ing a suitab le un iform morphism , where there is a constan t m such that | α ( a ) | = m for all letter s a . Define α : A ∗ 4 → A ∗ 4 by α (0) = 010301 210323 , α (1) = 010301 230323 , α (2) = 010301 232123 , α (3) = 010321 030123 , where the images ha ve length 12. For this we rely on the follo wing theorem by Crochemore [7]. Theor em 8 (Crochemor e) . A uniform morphism h : A ∗ → A ∗ is squa r e-fr ee if an d only if h pr eserves squar e-fr eeness of wor ds of length 3. One ca n de duce, or check by a computer , that α preserve s square- freeness of length three w ords. T herefo re, by Theorem 8, α ( w ) is square-fre e for all square- free infinite word s w over A 4 . Clearly , α ( w ) is C 4 -word . 3 T ourn ament words The abov e problem can be m odified for oriented graphs, i.e., directed graphs G = ( A n , E ) , where ij ∈ E implies j i / ∈ E . A tournamen t is an orientatio n of a complete graph. A word w ∈ A ∗ n is a tournament wor d , if for each diff erent i, j ∈ A n , if the word ij is a factor of w , then j i is not a facto r . The cas e first cas e of th e followin g result is obtained by a systemati c computer search . Theor em 9. (a) The long est squar e-fr ee tourna ment wor d over the four letter al- phabe t A 4 has length 20. These long est wor ds ar e w = 01201320 120320132032 and those obtain ed fr om w by permutin g the letters. (b) Ther e exi sts infinite squar e-fr ee tourna ment wor ds over A 5 . Pr oof. For the seco nd claim, consider the uni form morp hism α : A ∗ 3 → A ∗ 4 defined by α (0) = 012301 4 , α (1) = 013012 4 , α (2) = 012013 4 . The images α ( i ) ha ve length 7 . T he morphis m α can be easily seen to be squa re- free, sin ce the letter 4 occurs only at the end of the images , and in α ( i ) the letter is prece ded by the letter i . Hence if w is an infinite square -free word over A 3 , also α ( w ) is square-f ree, and clearl y it is a tourn ament word a G -wo rd over A 5 . 6 Refer ences [1] Noga Alon, Jarosła w Grytczuk, Mariusz Hałuszczak, and O li ver Riordan. Nonrepet iti ve colori ngs of graphs. Random Str uctur es A lgorith m s , 21(3- 4):336 –346, 2002. Random structures and algorithms (Poznan, 2001). [2] J ´ anos Bar ´ at and P ´ eter P . V arj ´ u. On square-free verte x colorings of graphs . Studia Sci. Math. Hungar . , 44(3):411 –422, 2007. [3] J ´ anos Bar ´ at and David R. W ood. Notes on nonrepet iti ve graph colouring. Electr on. J . 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