Choosability of a weighted path and free-choosability of a cycle

CHOOSABILI TY OF A WE I GHTED P A TH AND FREE-CHO O SABILITY OF A CYC LE YVES AUBR Y, J EAN-CHRIS TOPHE GODIN AND OLI V IER TOGNI Abstra ct. A graph G with a list of colors L ( v ) and w eigh t w ( v ) for eac h vertex v is ( L, w ) -colorable if one can choose a subset of w ( v ) colors from L ( v ) for each vertex v , such that adjacent v ertices receive disjoint color sets. In this paper, w e give necessary and sufficient cond itions for a w eig hted path to b e ( L, w )-colorable for some list as signments L . F urthermore, we solv e the problem of the free-choosabilit y of a cy cle. 1. Introduction The concept of c ho osabilit y of a graph, also calle d list coloring, has b een in tro duced by Vizi ng in [16], and indep endent ly b y Erd ˝ os, Ru bin and T aylo r in [5]. It con tains of course the colorabilit y as a particular case. Since its introd u ction, c ho osabilit y has b een extensiv ely stud ied (see f or example [1, 3, 14, 1 5 , 7 ] and more rec en tly [8, 9 ]). E v en for the original (un we igh ted) v ersion, the problem pro v es to b e difficult, and is NP-complete f or v ery restricted graph classes. Existing results f or the w eigh ted v ersion mainly concern the case of co nstan t w eights (i.e. ( a, b )-c h o osabilit y), see [1, 5, 8, 15]. F or the coloring problem of we igh ted graphs, quite a litt le bit more is kn own, see [13, 10, 11, 12]. This pap er consid ers list c olorings of w eigh ted graph s by stud ying cond i- tions on the list assignmen t for a wei gh ted path to b e c h o osable. Starting from the idea that in a path, the lists of colors of non consecutiv e v ertices do n ot in terfere, and follo win g the w ork in [6 ], w e in tro duce here th e notion of a wat erfall list assignm ent of a weig hted path. It is a li st assignm ent s uc h that any colo r is pr esen t on ly on one list or on t wo lists of consecutiv e ve r- tices. W e sho w that any list assignmen t (with some additional prop erties) can b e trans formed into a similar w aterfall list assignmen t. Then, using the result of Cropp er et al. [4] ab out Hall’s cond ition for list multico loring, w e pro v e a necessary and su ffi cien t co ndition for a w eigh ted path with a giv en waterfall list L to b e ( L, w )-colorable (Theorem 9) and use it to deriv e ( L, w )-colorabilit y resu lts for some general lists assignmen ts. In 1996, V oigt considered the follo wing pr oblem: let G b e a graph and L a list assignmen t and assume t hat an arb itrary v ertex v ∈ V ( G ) is pr ecolored b y a color f ∈ L ( v ). Is it alwa ys p ossible to complete this precoloring to a prop er list coloring ? Th is question lea ds to the concept of fr ee-c ho osabilit y in tro duced by V oigt in [17]. Date : Nov ember 22, 2021. 2010 Mathematics Subje ct Classific ation. 05C15, 05C38, 05C72. Key wor ds and phr ases. Coloring, Choosability , F ree-choosability , Cycles. 1 2 YVES A U BR Y, JEAN-CHRISTOPHE GODIN AND OLIVIER TOGNI W e in v estigate here th e free-c ho osabilit y of the fir st in teresting case, namely the cycle. As an app lication o f Theorem 9 , w e p ro ve our second main result which gives a necessary and su ffi cien t condition f or a cycle to b e ( a, b )-free-c ho osable (Theorem 12). In order to get a concise statemen t, w e in tro du ce the free-c h oice r atio of a graph, in the same wa y that Alon, T uza and V oigt in [1] in tr o duced th e c hoice ratio (wh ic h equals th e so-called fractional c h romatic num b er). In addition to the results obtained in this p ap er, th e study of w aterfall lists ma y b e of more general interest. F or no w on, the metho d is extended in [2] to b e used in a redu ction pr o cess, allo w in g to prov e colorabilit y r esults on triangle-free induced su b graphs of the triangular lattice . W e recall in Section 2 some definitions related to c h o osabilit y and fr ee- c ho osabilit y and introd uce the d efi nitions of the simila rit y b et ween t wo lists and of a waterfall list that are fundamental for this pap er. In Section 3, w e sho w ho w to transform a list into a similar w aterfall list and presen t a necessary and sufficien t condition for a w eigthed p ath to b e choosable. Theses result are u sed in Section 4 to obtain conditions for the ( L, w )- colorabilit y of a w eigh ted path and for the ( a, b )-free-c ho osabilit y of a cycl e. 2. Definitions and Preliminaries Let G = ( V ( G ) , E ( G )) b e a graph wh ere V ( G ) is the set of v ertices and E ( G ) is the set of edges, and let a , b , n and e b e inte gers. Let w b e a weigh t fun ction of G i.e. a map w : V ( G ) → N and let L b e a list assignment of G i.e. a map L : V ( G ) → P ( N ). By abuse of language and to simplify , we will ju st call L a list. If A is a finite set, w e denote by | A | the cardinal of A . A w eight ed graph ( G, w ) is a graph G together with a w eigh t fu nction w of G . Let us recall the definitions of an ( L, w )-colorable graph and an ( a, b )- free-c ho osable graph whic h are essent ial in this pap er. Definition 1. An ( L, w ) -c oloring c of a gr aph G is a map that asso ciate to e ach vertex v exactly w ( v ) c olors fr om L ( v ) such that adjac e nt vertic es r e c eive disjoints c olor sets, i.e. for al l v ∈ V ( G ) : c ( v ) ⊂ L ( v ) , | c ( v ) | = w ( v ) , and for al l v v ′ ∈ E ( G ) : c ( v ) ∩ c ( v ′ ) = ∅ . We say that G is ( L, w ) -c olor able if ther e exists an ( L, w ) -c oloring c of G . P articular cases of ( L, w )-colorabilit y are of grea t int erest. In ord er to in tro duce them, we define ( L, b )-co lorings and a -lists. An ( L, b )-coloring c of G is an ( L, w )-colo ring of G suc h that for all v ∈ V ( G ), w e ha ve w ( v ) = b. A a -list L of G is a list of G suc h that for all v ∈ V ( G ), we h av e | L ( v ) | = a. Definition 2. G is said to b e ( a, b ) -cho osable if for any a - list L of G , ther e exists an ( L, b ) -c oloring c of G . CHOOSABILITY OF P A THS AND FREE-CHOOSABILITY OF CYCLES 3 Definition 3. G is sa i d to b e ( a, b ) -fr e e-cho osable if for an y v 0 ∈ V ( G ) , and for any list L of G such that for any v ∈ V ( G ) \ { v 0 } , we have | L ( v ) | = a and | L ( v 0 ) | = b , ther e exists an ( L, b ) -c oloring c of G . W e d efine no w the similarit y of tw o lists with r esp ect to a w eighte d graph: Definition 4. L et ( G, w ) b e a weighte d gr aph. Two lists L and L ′ ar e said to b e sim ilar if this assertion is true: G is ( L, w ) -c olor able ⇔ G is ( L ′ , w ) -c olor able . The path P n +1 of length n is the graph with v ertex set V = { v 0 , v 1 , . . . , v n } and edge set E = S n − 1 i =0 { v i v i +1 } . T o simplify the n otations, L ( i ) denotes L ( v i ) and c ( i ) denotes c ( v i ). By analogy with the flo w of w ater in waterfalls, w e defin e a waterfall list as follo ws: Definition 5. A wa terfall list L of a p ath P n +1 of length n is a list L such that for al l i, j ∈ { 0 , . . . , n } with | i − j | ≥ 2 , we have L ( i ) ∩ L ( j ) = ∅ . Notice that another similar d efinition of a wate rfall list is that any color is pr esent only on one list or on tw o lists of consecutiv e v ertices. Figure 1 sho w s a list L of t he path P 5 (on the left), tog ether with a simila r w aterfall list L c (on the righ t). Definition 6. F or a weighte d p ath ( P n +1 , w ) , • A list L is goo d if | L ( i ) | ≥ w ( i ) + w ( i + 1) for any i, 1 ≤ i ≤ n − 1 . • Th e amplitude A ( i, j )( L ) (or A ( i, j ) ) of a list L is A ( i, j )( L ) = ∪ j k = i L ( k ) . L () 0 1 2 3 4 ✲ ✛ similar L c () 0 1 2 3 4 Fig. 1. Examp le of a list L whic h is similar to a wa terfall list L c . In [4], Cropp er et al. consid er Philip Hall’s th eorem on systems of distinct represent ativ es and its imp ro vemen t b y Halmos and V aughan as statemen ts ab out the existence of prop er list co lorings or list multic olorings o f c om- plete graphs. Th e n ecessary and s ufficien t c ondition in these theorems is 4 YVES A U BR Y, JEAN-CHRISTOPHE GODIN AND OLIVIER TOGNI generalized in the new setting as ”Hall’s condition” : ∀ H ⊂ G, X k ∈ C α ( H , L, k ) ≥ X v ∈ V ( H ) w ( v ) , where C = S v ∈ V ( H ) L ( v ) and α ( H, L, k ) is the indep end ence num b er o f the subgraph of H induced b y the v ertices con taining k in their color list. Notice that H can restricted to b e a connected ind uced subgraph of G . It is easily seen that Hall’s condition is n ecessary for a graph to b e ( L, w )- colorable. Cropp er et al. show ed that the condition is also su fficien t for some graphs, including paths: Theorem 7 ([4]) . F or the f ol lowing gr aphs, Hal l’s c ondition is sufficie nt to ensur e an ( L, w ) -c oloring: (a) cliques; (b) two cliques joine d by a c u t-vertex; (c) p aths; (d) a triangle with a p ath of length two adde d at one of its vertic es; (e) a triangle with an e dge adde d at two of its thr e e vertic es. This result is v ery nice , ho w ev er, it is often h ard to co mpute the left part of Hall’s condition, ev en for paths. Hence, for our study on c ho osabilit y of w eigthed paths, w e fi n d conv enien t to work w ith wa terfall lists for whic h , as w e will see in the next section, Hall’s condition is very easy to c hec k. 3. w a terf all lists W e first sho w that an y go o d list can b e transformed into a similar wa terfall list. Prop osition 8. F or any go o d list L of P n +1 , ther e exists a sim ilar wa terfal l list L c with | L c ( i ) | = | L ( i ) | for al l i ∈ { 0 , . . . , n } . Pr o of. W e are going to transform a go o d list L of P n +1 in to a w aterfall list L c and w e will pro v e th at L c is similar with L . First, remark that if a color x ∈ L ( i − 1) bu t x 6∈ L ( i ) for some i with 1 ≤ i ≤ n − 1, then for an y j > i , one can c hange the color x by a n ew color y 6∈ A (0 , n )( L ) in the list L ( j ), without c hanging the c ho osabilit y of the list. With th is remark in hand , w e can assume that L is suc h that an y color x app ears on the lists of consecutiv e vertic es i x , . . . , j x . No w, b y p erm uting the colors if necessary , w e can assum e that if x < y then i x < i y or i x = i y and j x ≤ j y . Rep eat the follo wing transformation: 1. T ak e the minim um col or x for whic h j x ≥ i x + 2 i.e. the color x is present on at least three v ertices i x , i x + 1 , i x + 2 , . . . , j x ; 2. Replace color x by a new colo r y in lists L ( i x + 2) , . . . , L ( j x ); unt il the obtained list is a w aterfall list (obvio usly , the num b er of iterations is alw a y s fin ite). No w, w e sh o w that this tran s formation preserv es the c ho osabilit y of the list. Let L ′ b e the list obtained fr om th e list L by the ab o ve transformation. If c is an ( L, w )-coloring of P n +1 then t he c oloring c ′ obtained from c b y c han ging the color x b y the color y in the colo r set c ( k ) of eac h ve rtex k ≥ i x + 2 (con taining x ) is an ( L ′ , w )-coloring since y is a new color. CHOOSABILITY OF P A THS AND FREE-CHOOSABILITY OF CYCLES 5 Con versely , if c ′ is an ( L ′ , w )-coloring of P n +1 , we co n sider t wo cases: Case 1 : x 6∈ c ′ ( i x + 1) or y 6∈ c ′ ( i x + 2). In this case, the co loring c obtained from c ′ b y c hanging the color y by t he color x in the color set c ′ ( k ) of eac h v ertex k ≥ i x + 2 (con taining y ) is an ( L, w )-coloring. Case 2 : x ∈ c ′ ( i x + 1) an d y ∈ c ′ ( i x + 2). W e hav e to consider t wo su b cases: • Sub case 1: L ′ ( i x + 1) 6⊂ ( c ′ ( i x ) ∪ c ′ ( i x + 1) ∪ c ′ ( i x + 2)). Th ere exists z ∈ L ′ ( i x + 1) \ ( c ′ ( i x ) ∪ c ′ ( i x + 1) ∪ c ′ ( i x + 2)) and the coloring c obtained from c ′ b y c h anging the colo r x b y the co lor z in c ′ ( i x + 1) and rep lacing the color y b y the color x in the color s et c ′ ( k ) of eac h v ertex k ≥ i x + 2 (con taining y ) is an ( L, w )-coloring. • Sub case 2 : L ′ ( i x + 1) ⊂ ( c ′ ( i x ) ∪ c ′ ( i x + 1) ∪ c ′ ( i x + 2)). W e hav e | L ′ ( i x + 1) | =     ( c ′ ( i x ) ∪ c ′ ( i x + 1) ∪ c ′ ( i x + 2)  ∩ L ′ ( i x + 1)    . As c ′ is an ( L ′ , w )-coloring of P n +1 , we ha v e | L ′ ( i x +1) | = | c ′ ( i x +2) ∩ L ′ ( i x +1) | + | c ′ ( i x +1) ∩ L ′ ( i x +1) | +     c ′ ( i x ) \ c ′ ( i x +2)  ∩ L ′ ( i x +1)    , | L ′ ( i x + 1) |− w ( i x + 1) − | c ′ ( i x + 2) ∩ L ′ ( i x + 1) | =     c ′ ( i x ) \ c ′ ( i x + 2)  ∩ L ′ ( i x + 1)    . Since y ∈ c ′ ( i x + 2) and y / ∈ L ′ ( i x + 1), w e obtain that | c ′ ( i x + 2) ∩ L ′ ( i x + 1) | ≤ w ( i x + 2) − 1 , hence  | L ′ ( i x + 1) | − w ( i x + 1) − w ( i x + 2)  + 1 ≤     c ′ ( i x ) \ c ′ ( i x + 2)  ∩ L ′ ( i x + 1)    . But, by hypothesis, L is a go o d list. Th u s | L ( i x + 1) | = | L ′ ( i x + 1) | ≥ w ( i x + 1) + w ( i x + 2) and 1 ≤     c ′ ( i x ) \ c ′ ( i x + 2)  ∩ L ′ ( i x + 1)    . Consequent ly , ther e exists z ∈  c ′ ( i x ) \ c ′ ( i x + 2)  ∩ L ′ ( i x + 1). Th e coloring c is t hen constructed from c ′ b y c hanging th e color x b y the color z in c ′ ( i x + 1), the color z b y the color x in c ′ ( i x ) and the color y by t he color x in the set c ′ ( k ) of eac h v ertex k ≥ i x + 2.  The follo wing theorem, wh ic h is a corol lary of Theorem 7, giv es a nec- essary and suffi cien t cond ition for a weigh ted path to b e ( L c , w )-colorable where L c is a w aterfall list. Theorem 9. L et L c b e a waterfal l list of a weighte d p ath ( P n +1 , w ) . Then P n +1 is ( L c , w ) -c olor able if and only if: ∀ i, j ∈ { 0 , . . . , n } , | j [ k = i L c ( k ) | ≥ j X k = i w ( k ) . 6 YVES A U BR Y, JEAN-CHRISTOPHE GODIN AND OLIVIER TOGNI Pr o of. “if ” p art: Recall that A ( i, j ) = ∪ j k = i L c ( k ). F or i, j ∈ { 0 , . . . , n } , let P i,j b e the subp ath of P n +1 induced b y th e vertic es i, . . . , j . By Theorem 7 , it is sufficien t to sho w that ∀ i, j ∈ { 0 , . . . , n } , X x ∈ A ( i,j ) α ( P i,j , L c , x ) ≥ j X k = i w ( k ) . Since the list is a waterfall list, then for eac h color x ∈ A ( i, j ), α ( P i,j , L c , x ) = 1 and th us P x ∈ A ( i,j ) α ( P i,j , L c , x ) = | A ( i, j ) | = | S j k = i L c ( k ) | . “only if ” part: If c is a ( L c , w )-coloring of P n +1 then ∀ i, j ∈ { 0 , . . . , n } : j [ k = i L c ( k ) ⊃ j [ k = i c ( k ) . Since L c is a waterfall list, it is easily seen that | S j k = i c ( k ) | = P j k = i w ( k ). Therefore, ∀ i, j ∈ { 0 , . . . , n } : | S j k = i L c ( k ) | ≥ P j k = i w ( k ).  4. Choosability of a p a th a nd fr e e-choosab ility of a cycle Theorem 9 has the follo win g corollary w h en the list is a goo d w aterfall list and | L ( n ) | ≥ w ( n ). Corollary 10. L et L c b e a waterfal l list of a weighte d p ath ( P n +1 , w ) such that for any i, 1 ≤ i ≤ n − 1 , | L c ( i ) | ≥ w ( i ) + w ( i + 1) and | L c ( n ) | ≥ w ( n ) . Then P n +1 is ( L c , w ) -c olor able if and only if ∀ j ∈ { 0 , . . . , n } , | j [ k =0 L c ( k ) | ≥ j X k =0 w ( k ) . Pr o of. Under the hyp othesis, if P n +1 is ( L c , w )-colorable, then Theorem 9 pro ves in particular the result. Con versely , since L c is a w aterfall list of P n +1 , we ha v e: ∀ i, j ∈ { 1 , . . . , n } , | A ( i, j ) | = |∪ j k = i L c ( k ) | ≥ |∪ j k = i k − i even L c ( k ) | = j X k = i k − i eve n | L c ( k ) | . Since L c is a go o d list of P n +1 (for simplicit y , w e s et w ( n + 1) = 0): ∀ i, j ∈ { 1 , . . . , n } , j X k = i k − i eve n | L c ( k ) | ≥ j X k = i k − i eve n ( w ( k ) + w ( k + 1)) ≥ j X k = i w ( k ) , then w e obtain for all i, j ∈ { 1 , . . . , n } , | A ( i, j ) | ≥ P j k = i w ( k ). S in ce f or all j ∈ { 0 , . . . , n } , | A (0 , j ) | ≥ P j k =0 w ( k ), Theorem 9 concludes the pro of.  Another in teresting coroll ary holds for lists L such that | L (0) | = | L ( n ) | = b , and fo r all i ∈ { 1 , . . . , n − 1 } , | L ( i ) | = a . The function Ev en is defined for an y real x b y: Ev en( x ) is the smallest ev en integ er p suc h that p ≥ x . Corollary 11. L et L b e a list of P n +1 such that | L (0 ) | = | L ( n ) | = b , and | L ( i ) | = a = 2 b + e for al l i ∈ { 1 , . . . , n − 1 } (with e 6 = 0 ). CHOOSABILITY OF P A THS AND FREE-CHOOSABILITY OF CYCLES 7 If n ≥ Ev en  2 b e  then P n +1 is ( L, b ) -c olor able. Pr o of. The hyp othesis implies th at L is a go o d list of P n +1 , h ence by Prop o- sition 8, there exists a wa terfall list L c similar to L . So w e get: ∀ i ∈ { 1 , . . . , n − 1 } , | L c ( i ) | ≥ 2 b = w ( i ) + w ( i + 1) and | L c ( n ) | ≥ b = w ( n ). By Corollary 10 it remains to pro ve that: ∀ j ∈ { 0 , . . . , n } , | A (0 , j ) | ≥ j X k =0 w ( k ) = ( j + 1) b. Case 1 : j = 0. By hyp othesis, w e h a ve | A (0 , 0) | = | L c (0) | ≥ b . Case 2 : j ∈ { 1 , . . . , n − 1 } . Since L c is a waterfall list of P n +1 w e obtain that: if j is ev en | A (0 , j ) | ≥ j X k =0 k even | L c ( k ) | = b + j X k =2 k even 2 b = b + j 2 2 b = ( j + 1) b, and if j is o dd | A (0 , j ) | ≥ j X k =0 k odd | L c ( k ) | = j X k =1 k odd 2 b = j + 1 2 2 b = ( j + 1) b. Hence for all j ∈ { 0 , . . . , n − 1 } , | A (0 , j ) | ≥ ( j + 1) b . Case 3 : j = n . Since n ≥ Ev en  2 b e  b y hyp othesis, and | A (0 , n ) |≥ n X k =0 k od d | L c ( k ) | =  a n 2 if n is ev en b + a n − 1 2 other w ise w e deduce that | A (0 , n ) | ≥ ( n + 1) b , whic h concludes the pro of.  F or example, let P n +1 b e the path of length n w ith a list L su c h that | L (0) | = | L ( n ) | = 4, and | L ( i ) | = 9 for all i ∈ { 1 , . . . , n − 1 } . Then the previous Corollary tells us th at we can fi nd an ( L, 4)-coloring of P n +1 when- ev er n ≥ 8 . In other words, if n ≥ 8, we can c ho ose 4 colors on eac h v ertex suc h that adjacen t vertic es r eceiv e d isj oin t colors. If | L ( i ) | = 11 for all i ∈ { 1 , . . . , n − 1 } , then P n +1 is ( L, 4)-colorable wheneve r n ≥ 4. The ab ov e result is a starting to ol used in [2 ] to attac k McD iarmid and Reed’s conjecture clai ming that ev ery triangle free induced subgraph of the triangular lattice is (9 , 4)-col orable (hence the v alues a = 9 and b = 4 are someho w “natural”). It is also u sed in the follo wing to d etermine the fr ee- c hoice-ratio of the cycle. The cyc le C n of length n is the graph with vertex set V = { v 0 , . . . , v n − 1 } and edge set E = S n − 1 i =0 { v i v i +1( mod n ) } . 8 YVES A U BR Y, JEAN-CHRISTOPHE GODIN AND OLIVIER TOGNI Let F C h ( x ) b e the set of graph s G whic h are ( a, b )-free-c h o osable for all a, b su c h that a b ≥ x : F C h ( x ) = { G | ∀ a b ≥ x, G is ( a, b )-free-c ho osable } . Moreo v er, we can define the free-c hoice ratio fc hr( G ) of a graph G b y: fc hr ( G ) := inf { a b | G is ( a, b )-free-c ho osable } . If ⌊ x ⌋ denotes the greatest in teger less or equal to the real x , w e can state: Theorem 12. If C n is a cycle of length n , then C n ∈ F C h (2 + j n 2 k − 1 ) . Mor e over, we have: fc hr ( C n ) = 2 + j n 2 k − 1 . Pr o of. Let a, b be tw o integ er s suc h that a/b ≥ 2 + ⌊ n 2 ⌋ − 1 . Let C n b e a cycle of length n and L a a -list of C n . Without loss of generalit y , w e can supp ose that v 0 is the ve rtex c hosen for the fr ee-c ho osabilit y a nd L 0 ⊂ L ( v 0 ) has b elemen ts. It remains to construct an ( L, b )-coloring c of C n suc h that c ( v 0 ) = L 0 . Hence we ha ve to construct an ( L ′ , b )-coloring c of P n +1 suc h that L ′ (0) = L ′ ( n ) = L 0 and f or all i ∈ { 1 , ..., n − 1 } , L ′ ( i ) = L ( v i ). W e ha ve | L ′ (0) | = | L ′ ( n ) | = b and for all i ∈ { 1 , ..., n − 1 } , | L ′ ( i ) | = a . Since a/b ≥ 2 + ⌊ n 2 ⌋ − 1 and e = a − 2 b , w e get e/b ≥ ⌊ n 2 ⌋ − 1 hence n ≥ Ev en(2 b/e ). Using Corollary 11, we ge t: C n ∈ F C h (2 + j n 2 k − 1 ) . Hence, w e h a ve that fc hr ( C n ) ≤ 2 + ⌊ n 2 ⌋ − 1 . Moreo v er, let us p ro ve that M = 2 + ⌊ n 2 ⌋ − 1 is reac hed. F or n o d d, V oigt has pr o ve d in [18] that th e c hoice ratio c hr( C n ) of a cycle of o dd length n is exact ly M . Hence fchr( C n ) ≥ chr( C n ) = M , and the result is prov ed . F or n ev en, le t a, b b e tw o in tegers such that a b < M . W e construct a coun terexample for the free-c ho osabilit y : let L b e the list of C n suc h that L ( i ) =        { 1 , . . . , a } if i ∈ { 0 , 1 } { 1 + i − 1 2 a, . . . , ( i − 1 2 + 1) a } if i 6 = n − 1 is odd { b + 1 + i − 2 2 a, . . . , b + ( i − 2 2 + 1) a } if i is ev en an d i 6 = 0 { 1 , . . . , b, 1 + ( n − 4 2 + 1) a, . . . , ( n − 4 2 + 2) a − b } if i = n − 1 If we choose c 0 = { 1 , . . . , b } ⊂ L (0), w e can c heck that it do es not exist an ( L, b )-coloring of C n suc h that c (0) = c 0 , so we could not do b etter.  Remark 13. In p articular, the pr evi ous the or e m implies that if n ≥ Ev en ( 2 b e ) then the cyc le C n of length n is (2 b + e, b ) -fr e e- cho osable. CHOOSABILITY OF P A THS AND FREE-CHOOSABILITY OF CYCLES 9 Remark 14. Er d˝ os, R ub in and T aylor ha ve sta te d in [5] the fol lowing ques- tion: If G is ( a, b ) -c olor able, and c d > a b , do es it imply that G is ( c, d ) - c olor able ? Gutner and T arsi have shown in [8 ] that the answer is ne gative in gener al. If we c onsider the analo gue question for fr e e-cho osability, then the pr evious the or em implies that it is true for the cycle. Referen ces [1] N. A lon, Zs. T uza, M. V oigt, Cho osabilit y and fractional chromatic num b er , Discrete Math. 165/166, ( 1997), 31-38. [2] Y. Aubry , J.-C. Godin and O. T ogni, Extended core and choosabilit y of a graph , arXiv:1006.29 58 v1 [cs.DM], 2010. [3] O.V. Boro din, A.V. Kosto chk a, D.R. W o od all, List edge and list colourings o f multi- graph , J. Combin. Theory Series B, 71 : 184-204, (1997). [4] M. M. Cropp er, J. L. Goldw asser, A. J. W. Hilton, D. G. Hoffman, P . D. Johnson, Extending the disjoin t-representative s theorems of Hall, Halmos, and V aughan to list- multico lorings of graph s. J . Graph Theory 33 (2000), no. 4, 199 ¯ D219. [5] P . Erd ˝ os, A.L Rub in and H. T aylor, C ho osabilit y in graphs , Proc. W est-Coast Conf. on Combinatori cs, Graph Theory and Computing, Congressus N umerantium XXVI, (1979), 125-157. [6] J.-C. Go din, Coloration et c hoisissabilit ´ e des graphes et applications , PhD thesis (i n frenc h), Universit ´ e d u Su d T oulon-V ar, F rance (2009). [7] S. Gravier, A Ha j´ os-like theorem for list coloring , Discrete Math. 152, ( 1996), 299-302. [8] S. Gutn er and M. T arsi, Some results on (a:b)-c ho osability , Discrete Math. 309, (2009), 2260-2270 . [9] F. Hav et, C h oosability of the square of planar su b cubic graphs with large girth . D is- crete Math. 3 09, (2009), 3553-3563. [10] F. H a vet, Channel assignemen t and m ulticolouring of the indu ced su bgraphs of the triangular lattice . Discrete Math. 2 33, (2001), 219-233. [11] F. Hav et, J. Zerovnik. Finding a five bicolouring of a triangle-free subgraph of the triangular lattice . Discrete mathematics 244, (2002), 103-108. [12] M. Kchik ech and O. T ogni. A pproximation algorithms for multicol oring p ow ers of square and triangular meshes , Discrete Math. and Theoretical Computer Science, V ol. 8 (1):159-172, 20 06. [13] C. McDiarmid and B. R eed. Channel assignement and w eighted coloring. Netw orks, 36, (2000), 114-117. [14] C. Thomassen, The chromatic num b er of a graph of girth 5 on a fixed su rface , J. Com b in. Theory , (2003), 38-71. [15] Zs. T uza and M. V oigt, Every 2-choosable graph is (2m,m)-choosable , J. Graph The- ory 22, (1996), 245-252. [16] V. G Vizing, Coloring the vertices of a graph in p rescribed colors (in Russian) , Diskret. Analiz. No. 29, Meto dy Diskret. Anal. v T eorii Ko dov i Sh em 101 (1976), 3-10. [17] M. V oigt, Cho osabilit y of planar graphs , Discrete Math., 150, (1996), 4 57-460. [18] M. V oigt, On list Colourings and Cho osabilit y of Graphs , A bilitationssc hrift, TU Ilmenau (1998). Institut de M a th ´ ema tiques de Toulon, Universit ´ e d u Sud Toulo n-V ar, France, and Labora toire LE2I, U n iversit ´ e de Bour gogne, France E-mail addr ess : yves.aub ry@univ-tl n.fr, godinjean chri@yahoo. fr and olivier.togni@u- bourgogne.f r