Rocheste Institute of Technology RIT Schola Wos Theses Thesis/Dissetation Collections 8--08 The Chomatic Villainy of Complete Multipatite Gaphs Anna Raleigh an9@it.edu Follow this and additional wos at: https://scholawos.it.edu/theses Recommended Citation Raleigh, Anna, "The Chomatic Villainy of Complete Multipatite Gaphs" (08). Thesis. Rocheste Institute of Technology. Accessed fom This Thesis is bought to you fo fee and open access by the Thesis/Dissetation Collections at RIT Schola Wos. It has been accepted fo inclusion in Theses by an authoized administato of RIT Schola Wos. Fo moe infomation, please contact itscholawos@it.edu.
The Chomatic Villainy of Complete Multipatite Gaphs by Anna Raleigh A Thesis Submitted in Patial Fulfillment of the Requiements fo the Degee of Maste of Science in Applied Mathematics School of Mathematical Sciences, College of Science Rocheste Institute of Technology Rocheste, NY August nd, 08
Committee Appoval: D. Paul Wenge August, 08 Rocheste Institute of Technology Thesis Adviso D. Jobby Jacob August, 08 Rocheste Institute of Technology Committee Membe D. Daen Naayan August, 08 Rocheste Institute of Technology Committee Membe D. Sogol Jahanbeam August, 08 San Jose State Univesity Committee Membe
Chomatic Villainy D. Matthew Hoffman August, 08 School of Mathematical Sciences Diecto of Gaduate Pogams
Chomatic Villainy Abstact Suppose the colos in a χ(g)-coloing of a gaph G have been eaanged. We will call this eaangement c. The chomatic villainy of the c is defined as the minimum numbe of vetices that need to be ecoloed in ode to etun c to a pope coloing in which each colo appeas the same numbe of times as in the initial coloing. The maximum chomatic villainy when consideing all eaangements of all χ(g)-coloing of G is the chomatic villainy of G. Hee, the chomatic villainies of cetain families of gaphs wee investigated and the chomatic villainies of paths and cetain classes of complete multipatite gaphs wee found. Bounds wee found fo cetain classes of odd cycles and complete multipatite gaphs as well. i
Chomatic Villainy Contents I Intoduction I. Bacgound.......................................... I. Tems.............................................. II Complete Multipatite Gaphs 8 III Paths 9 IV Odd Cycles IV. Odd Cycles with one vetex of colo........................... IV. Odd cycles with two vetices of colo.......................... V Conclusions and Open Questions 6 VI Bibliogaphy 8 ii
Chomatic Villainy I. Intoduction I. Bacgound A gaph G is defined as an odeed pai (V, E) whee V(G) epesents the vetex set of G and E(G) epesents the set of edges connecting the vetices of G. An edge between two vetices x and y is notated as xy o yx. Two vetices of G ae adjacent o neighbos if an edge exists between them. The neighbohood of x, denoted N(x), is the set of vetices adjacent to a vetex x. A pope -coloing of G is a labeling of the vetices of G using colos, usually epesented by numbes though, in such a way that adjacent vetices eceive diffeent colos. The chomatic numbe of a gaph G, denoted χ(g), is the smallest such that a pope -coloing of G exists. We define c(v j ) as the colo of vetex v j unde coloing c. Many algoithms exist to poduce pope coloings. These algoithms also give uppe bounds on the chomatic numbe of a given gaph. Howeve, these algoithms do not always esult in a χ(g)-coloing of G. One such algoithm is efeed to as geedy coloing. In a geedy coloing, the vetices of G ae given an ode v, v,..., v V(G). Colos ae assigned by going though the vetices in ode and assigning the colo with the lowest numbe that is not given to a peviously coloed adjacent vetex. The odes given to the vetices of G changes the uppe bound on χ(g). Note that thee is an odeing fo evey gaph that will use the chomatic numbe of colos when fed though the geedy algoithm. Howeve, thee is a gaph G (in fact thee ae many) with chomatic numbe that when put into the geedy coloing with the wong odeing will have a chomatic numbe that will gow with log ( V(G) ). Fo example, conside the gaph given in Figue. It is popely coloed with two colos. If we wee to implement a geedy coloing in which we coloed the vetices in ode fom left to ight, we would achieve the coloing given in Figue. While this coloing is pope, it uses fou colos. Note that = log (8) + and this gaph has eight vetices. Figue : A pope coloing with colos.
Chomatic Villainy Figue : A pope coloing with colos. Let (G) be the maximum degee of a vetex in G and δ(g) be the minimum degee of a vetex in G. Given that a vetex in G has at most (G) neighbos, a geedy coloing will use at most (G) + colos. Thus, χ(g) (G) + [6]. Howeve, (G) + is aely a stict uppe bound. It was poven by Boos in [] that if G is a connected gaph othe than a complete gaph o an odd cycle, then χ(g) (G). A sticte bound on χ(g) was given by Welsh and Powell in [5]. Let d i be the degee of vetex i. Let the vetices in the geedy coloing be odeed in non-inceasing ode of degee, such that d d... d V(G). The colo given to the vetex with degee d i is at most one geate than the numbe of neighbos of d i that ae aleady coloed. This value is bounded by d i and i. Thus, a sticte bound on χ(g) is given by χ(g) + max i (min(d i, i )). In [], Szeees and Wilf found an uppe bound on χ(g) using the degees of the subgaphs of G to ode the vetices. This bound is given by χ(g) + max H G δ(h). Subgaphs and complete gaphs ae defined in section.. Cla et al. [] intoduced the concept of chomatic villainy. Let us assume that χ(g) = and let c be a pope -coloing of G. Let c be a coloing of G that is a eaangement of the colos in c. The wea chomatic villainy of c, denoted B w (c ), is the minimum numbe of vetices that must be ecoloed with the same set of colos as c in ode to e-obtain a pope coloing. The villainy of c, denoted B(c ), is the minimum numbe of vetices that must be ecoloed with the same set of colos as c with the additional stipulation that each colo must appea exactly as many times as it does in c. The wea villainy of the gaph G is the the lagest numbe of vetices that need to be ecoloed ove all eaangements of all χ(g)-coloings of G. Let c be a pope χ(g)-coloing of G and let c be a eaangement of c. The wea villainy of G is given by ( ) B w (G) = max max (B w (c )). c c The villainy of the gaph G is the lagest numbe of vetices that need to be ecoloed ove all
Chomatic Villainy eaangements of all -coloings of G whee each colo in the esulting pope coloing appeas exactly as many times as it does in c. Cla et al. [] established that B w (G) B(G) given that any ecoloing that follows the stipulations of villainy is also valid unde the stipulations of wea villainy. Let c be a pope χ(g)-coloing of G and let c be a eaangement of c. The villainy of G is given by ( ) B(G) = max max(b(c )). c c Conside the following example. Let G be a path on 7 vetices with optimal coloing c as shown in Figue and let c be the eaangement of c given by Figue. Figue : A popely coloed path on 7 vetices. Figue : An impopely coloed path on 7 vetices. In ode to popely colo an odd path, the colo that appeas most often must be on the outemost two vetices and the colos must altenate. In the case whee the numbe of each colo is maintained, the outemost vetices of G must eceive colo. Theefoe, to e-obtain a pope coloing of G while maintaining the numbe of each colo that appeaed in c, the coloing must be identical to c. This can be obtained by ecoloing the fist, thid, fouth, and sixth vetices. It holds that B(c ) =. In the wea case, ecoloing the second, fifth, and seventh vetices estoes G to a pope coloing. Howeve, this pope coloing is not equivalent to c. The colo appeas fou times in G while appeas thee times. Theefoe, the two oute-most vetices eceive colo. It holds that B w (c ) =. In [], Cla et al. poved seveal esults egading both chomatic villainy and wea chomatic villainy. It was detemined that B(G) = B w (G) = 0 if and only if G is a complete o empty gaph. Additionally, the class was detemined fo gaphs with a wea chomatic villainy of. Additional
Chomatic Villainy esults have been found egading the villainies of uniquely coloable and pseudo-uniquely coloable gaphs, as well as othe categoies of gaphs such as connected bipatite gaphs, cycles, disjoint unions of gaphs, and cetain classes of subgaphs. The authos also posed a numbe of open questions that can be summaized as follows: What ae the chaacteistics of gaphs with a chomatic villainy of? Is it the case that the chomatic villainy of a cycle with + vetices is when? What ae the chomatic villainies of complete multipatite gaphs? What ae the lagest possible values of B(G) and B w (G) when G has n vetices and χ(g) =? Is the wea chomatic villainy of the disjoint union of two gaphs geate than o equal to the sum of the two gaph s espective wea villainies? We will focus on the chomatic villainy of complete multipatite gaphs, paths, and odd cycles. I. Tems A gaph H is a subgaph of G, denoted H G, if V(H) V(G) and E(H) E(G). It holds that G contains its subgaphs. A gaph is complete if evey pai of vetices ae adjacent. Note that a complete gaph on n vetices is denoted K n. A matching in G is a set of edges in which no two edges shae a vetex. Edges that do not shae a vetex ae also efeed to as independent edges. The matching coves the vetices in its edges. A matching is pefect if it coves evey vetex in G. A gaph G is weighted if thee ae numeical values o weights assigned to its edges. G is unweighted othewise. Note that an independent set is a set of vetices in a gaph in which none ae adjacent. A gaph G is bipatite if V(G) consists of two independent sets. The maximum weighted matching in a bipatite gaph is a matching in which the sum of the edge weights has a maximal value. The value is maximal if it cannot be made lage. Let G be a bipatite gaph with independent sets X and Y. By Hall s Theoem, a matching exists that coves evey vetex in X iff fo evey S whee S X, N(S) S []. Any undefined tems can be found in [6]. Section II will focus on the chomatic villainy of complete multipatite gaphs. A gaph is -patite
Chomatic Villainy if it is the union of independent sets. These sets ae efeed to as patite sets. Note that a patite set can be empty. Theefoe, a -patite gaph is also j-patite fo all j. A -patite gaph can also be efeed to as multipatite. An example of an incomplete 5-patite gaph is given in Figue 5. Let P i be a patite set in a -patite gaph fo all i []. A -patite gaph with patite Figue 5: An incomplete 5-patite gaph sets P, P,...,P is consideed complete if fo all i, j {,,..., }, evey vetex in P i is adjacent to evey vetex in P j iff i = j. Such a gaph will be notated K n,n,...,n whee n i is the size of P i fo i {,,..., }. Let i be the numbe of patite sets of size n i. Without loss of geneality, we will notate a complete multipatite gaph with i= i patite sets as Kn,..., n }{{},n,..., n,...,n }{{},..., n }{{} such that n n n... n > 0. In the event that i n i = i+ n i+ fo some i {,,..., }, the set of lage patite sets will eceive the lowe index. That is, n i > n i+ when i n i = i+ n i+. Note that in geneal, n i need not be geate than n i+. Conside the example given in Figue 6. The gaph in Figue 6 is a complete 7-patite gaph with five patite sets of size and two patite sets of size. Theefoe, the gaph is denoted K,,,,,, }{{}}{{} 5 n =. and n = while Figue 6: A complete 7-patite gaph A gaph G is uniquely coloable if pope coloings of G using χ(g) colos diffe only by the names 5
Chomatic Villainy of the colos []. Note that a complete multipatite gaph is uniquely coloable; thee is only one way to colo a complete multipatite gaph up to pemutation of the colos. Given that evey vetex in a given patite set P i is adjacent to evey vetex in P j fo all i = j, each patite set must be coloed with a diffeent colo. Because each patite set is an independent set, a given patite set can be coloed with one colo. Note that the chomatic numbe of a gaph G whee G = Kn,..., n }{{},n,..., n,...,n }{{},..., n is j= }{{} j, the total numbe of patite sets. Fo example, a pope coloing c of K,,,, is given by Figue 7 and K,,,, has a chomatic numbe of 5. Conside a eaangement of c. To etun G to a pope coloing, evey colo that appeas n i times must be in a patite set of size n i fo i [5]. Note that in the pope ecoloing, the vetices in a patite set might have a diffeent colo than they did in c. Let c be the eaangement of c shown in Figue 8. Unde c, the patite sets of size contain colos that appea twice, and the patite set of size contains colos that appea once. Theefoe, evey vetex in these patite sets must be e-coloed. All of the vetices in the patite sets of size appea thee times. Howeve, the colo appeas moe often in the left patite set of size and the colo appeas moe often in the ight patite set of size. Theefoe, the esulting pope coloing when the smallest possible numbe of ecoloings ae pefomed is given by Figue 9. 5 Figue 7: A pope coloing c of K,,,, Note that Figue 9 is not the oiginal pope coloing. To obtain the oiginal pope coloing, eight ecoloings would be needed while this ecoloing was obtained in six ecoloings. Howeve, B(G) = 6. Let c be the eaangement of c given by Figue 0. In Figue 0, only two colos, and, ae in a patite set of the ight size, size. They cannot both be coect in the same patite set. Theefoe, only one vetex is coloed coectly in this eaangement and the coloing has a 6
Chomatic Villainy 5 Figue 8: An impope ecoloing of c 5 Figue 9: A pope ecoloing of K,,,, chomatic villainy of 9. Note that this is equivalent to i= n in i j= jn j. It holds that thee ae only fou vetices with colos that don t appea in the gaph thee times and six vetices with colos that appea thee times. Theefoe, unde any ecoloing of G, at least two of the vetices in the patite sets of size must have a colo that appeas in the gaph thee times. At least one of these vetices will be coloed coectly. Theefoe, the villainy of the gaph cannot be geate than 9. 5 Figue 0: An impope ecoloing of c 7
Chomatic Villainy II. Complete Multipatite Gaphs Let G = Kn,..., n }{{},n,..., n,...,n }{{},..., n be a complete multipatite gaph with i pats of size }{{} n i fo i [], whee n n... n > 0. Let P i be a patite set in G fo i [χ(g)] and let P, P,..., P be the patite sets of size n. The villainy of G is dependent on the numbe of vetices in χ(g) j= + P j in elation to the numbe of vetices in i= P i. It holds that χ(g) j= + P j contains i= in i vetices. Multipatite gaphs can be soted into thee cases:. n j= jn j,. n > j= jn j and j= jn j <. n > j= jn j and j= jn j ( n n ), and ( ) n n. Conside case whee i n i j =i j n j fo all i and all j [] whee j = i. It holds that no set of patite sets of the same size contains moe than vetices. Conside a consecutive labeling V(G) of the vetices whee each vetex eceives a label v l in G such that the vetices in the i patite sets of size n i eceive labels that pecede those in the i+ patite sets of size n i+. Fo example, in K,,,,,, P contains v, v, v, and v, P contains v 5, v 6, v 7, and v 8, P contains v 9, v 0, and v, P contains v, v, and v, P 5 contains v 5, and v 6, and P 6 contains v 7, and v 8. Let P i get colo i unde the pope χ(g)-coloing c. Since no set of patite sets of the same size compises moe than half the gaph, the colo of vetex v i unde c does not have the same colo as v i+ V(G) (mod V(G) ) (note that if i + V(G) (mod V(G) ) = 0, then v i+ V(G) (mod V(G) ) = v V(G) ). Futhemoe, c(v i ) and c(v i+ V(G) ) each appea a diffeent numbe of times (mod V(G) ) in G. Theefoe, if the colos of the vetices in G ae edistibuted such that v i+ V(G) (mod V(G) ) 8
Chomatic Villainy eceives colo c(v i ), no vetex in a set of size n i eceives a colo that appeas n i times in G fo all i []. Conside G = K,,,,,. A labeling and a pope coloing of K,,,,, ae given in Figue. In this case, = 9. The coloing achieved by ecoloing evey vetex v i with c(v i+9 (mod 8) ) is V(G) given in Figue. Note that in this coloing, no colo that appeas fou times is in a patite set of size, no colo that appeas thee times is in a patite set of size, and no colo that appeas twice is in a patite set of size. Theefoe, evey vetex in Figue must be ecoloed to achieve a pope coloing, and B(G) = 8. v v 8 7 v v v v 6 6 v 6 v 5 v 5 6 7 v 8 5 5 v v 9 v v 0 v v Figue : A pope coloing of K,,,,, 5 6 6 Figue : An impope ecoloing of K,,,,, with a villainy of V(G) Theoem. Let G = Kn,..., n }{{},n,..., n,...,n }{{},..., n be a complete multipatite gaph with i patite }{{} sets of size n i fo all i [] whee n n n > 0. If fo all i [] we have n j= jn j, then B(G) = V(G). 9
Chomatic Villainy Poof. Fo j i= i, let P j be a patite set in G. Note that χ(g) = i= i. Iteating though the patite sets such that the i sets of size n i pecede the i+ sets of size n i+ we label the vetices v, v,..., v V(G). That is, P = {v, v... v n }, P = {v n +, v n +,..., v n },. P χ(g) = { } v d= dn d +( )n +, v d= dn d +( )n +,..., v d=. dn d Let c be a pope coloing of V(G) that uses the smallest numbe of colos. Theefoe all vetices in each patite set get the same colo unde c. Let us define c such that the vetices in P i eceive colo i. The gaph G can be epesented as a cicle whee the vetices ae labeled consecutively. It holds that i n i V(G) fo all i []. Note that if a vetex has index 0 (mod V(G) ), it ( ) will ecieve label v V(G). Theefoe, c v l+ V(G) = c(v (mod V(G) ) l ). Futhemoe, c(v l ) and ( ) c each appea a diffeent numbe of times in G. v l+ V(G) (mod V(G) ) V(G) Let c be a clocwise otation of the colos of the vetices unde c by units. We define c (v l ) ) to be the colo of vetex l unde c. Fo all l [ V(G) ], it holds that c (v l+ V(G) = (mod V(G) ) ( ) c(v l ). Since c(v l ) and c each appea a diffeent numbe of times in G, v l+ V(G) (mod V(G) ) l [ V(G) ]. Thus, v l+ V(G) (mod V(G) ) must be ecoloed to etun G to a pope coloing. This holds fo all B(G) = V(G). Let G = Kn,..., n }{{},n,..., n,...,n }{{},..., n be a complete multipatite gaph such that n > }{{} j= jn j. Let c be a pope χ(g)-coloing of G whee the vetices in P i eceives colo i. In both case and case, any eaangement of c will have at least n j= jn j vetices in i= P i that have a colo in {,,..., }. If n, then ( ) n n will be equivalent to 0. The value j= jn j cannot be negative, and 0
Chomatic Villainy ( in case, j= jn j is stictly less than n n ) (. Theefoe, if n and 0 whee n j= jn j, the gaph falls into case. Note that the value n n n ) = epesents how many times one vetex of each colo in {,,..., } can be placed in each patite set of size n. In ( ) a given P i whee i [ ], n n vetices emain uncoloed afte coloing n vetices in ( ) P i with each colo in {,,..., }. Thus, the value n n is equivalent to how many vetices ae left uncoloed in G afte n in each patite set in {P, P,..., P }. Note that is stictly less than. Theefoe, if n vetices of each colo in {,,..., } have been placed ( ) n n is equivalent to n (mod ) and vetices in each patite set in i= ae coloed with each colo in {,,..., }, any emaining uncoloed vetices that need to be coloed with colos in {,,..., } can no longe be coloed such that each colo in {,,..., } is placed in a single patite set. In case whee n > j= jn j and j= jn j < i= P i such that n n ( ) n n, we fist colo the vetices in vetices in each patite set get each colo in {,,..., }. This esults in ecoloed vetices in i= P i. Let us efe to the uncoloed potion of P i as Q i. Note that each Q i contains n n vetices. ( ) It holds that n n i= in i vetices in G still need to be coloed with a colo in ( ) ( ) {,,..., }. By ou assumption, j= jn j < n n. Theefoe, n n ( ) i= in i > 0. These n n i= in i coloed vetices cannot be distibuted such that one of each colo in {,,..., } is in each Q i. We will distibute these colos such that n n vetices of each consecutive colo ae used and ae distibuted consecutively acoss Q i s. It holds that at least i= Q i. Theefoe, Subsequently, n n i= in i n n n n i= in i n n n n i= in i n n total colos in {,,..., } must be used in patite sets contain n + vetices of a unique colo. patite sets must be coloed with a colo that appeas
Chomatic Villainy n times. It follows that at most n n i= in i n n n n n + i= ( ) in i n n n + n n n = + i= in i n n vetices do not need to be ecoloed. Conside G = K,,,. A pope coloing of G is given in Figue such that the vetices in P i eceive colo i. In this case, ( n n ) ( = 8 while i= in i = and it holds that i= in i < ) 8 = 6 ( ) n n. We fist ecolo 8 vetices in P, P, and P with each colo in {,, } as shown in Figue. It holds that ( 8 8 ) = and V(Q i ) = fo all i {,, }. Note that at least n n i= in i (8) 8 n n = 8 8 = = total colos in {,,..., } must be used in i= Q i. Thus the emaining fou vetices that must be coloed with a colo in {,, } will be coloed such that Q and Q contain a vetex with colo and Q and Q contain a vetex of colo. The additional two vetices in i= P i will eceive colo and the emaining unused colos ae placed in P. This ecoloing is given in Figue 5. In the coloing given in Figue 5, thee ae at most vetices in a patite set that have the same colo. Howeve, thee ae only two distinct colos that appea thee times and the vetices in one patite set must be coloed with a colo that only appeas twice. Recoloing the vetices in P with would equie five ecoloings. Liewise, ecoloing the vetices in P with would equie five
Chomatic Villainy Figue : A pope coloing of K 8,8,8, Figue : A patial ecoloing of K 8,8,8, ecoloings. This leaves the vetices in P to be ecoloed with, equiing six ecoloings. Thus, to etun the gaph in Figue 5 to a pope coloing, 8 vetices need to be ecoloed and 8 ae aleady coectly coloed. Note that n n n + i= in i n n = 8 + (8) 8 8 = 8. 8 Theoem. Let G = Kn,..., n }{{},n,..., n,...,n }{{},..., n be a complete multipatite gaph with j patite }{{} sets of size n j fo all j. Note that n n... n > 0. If we have n > j n j and j= ( j n j < n j= n ),
Chomatic Villainy Figue 5: An impope ecoloing of K 8,8,8, then Poof. Fo j uses χ(g) colos. B(G) j n j j= n n n j= jn j n n. [ ] j= j, let P j be a patite set in G. Let c be a pope coloing of V(G) that Theefoe all vetices in each patite set get the same colo unde c, and χ(g) = i= colos ae used. We define c such that P, P,..., P ae the sets of size n and each vetex in P i whee i [ ] has colo i. Note that j= jn j is the total numbe of vetices in χ(g) i= + P i. n n j= jn j n, we define a coloing c n of G that is a eaangement of a χ(g)-coloing of G. Fo each i and j whee i {,..., } and j {,..., } To pove B(G) j= jn j n we colo n vetices in P i with colo j. This esults in each P i having n n vetices that ae not yet coloed. It holds that n n colo in {,,..., }. By ou assumption, j= jn j < of vetices in i= P i that must eceive a colo in {,,..., } is n j= j= jn j vetices in i= P i must still eceive a ( ) n n. Thus, the total numbe ( j n j > n n = n. n ) It follows that n n j= jn j is geate than 0.
Chomatic Villainy Let us efe to the uncoloed vetices in P i as Q i. We aim to colo these emaining vetices such that the fewest possible numbe of colos in {,..., } is used and these colos ae distibuted such that at most one of each colo appeas in each Q i. We will colo the vetices in i= Q i such that j= jn j vetices eceive colos fom the set { +,..., χ(g)} and the emaining n n j= jn j vetices, those that ae not yet coloed, eceive colos in {,..., }. So fa, each colo in {,..., } has been used n times in i= P i. Theefoe, since each colo in {,..., } appeas at most n times in G, each colo in {,..., } can appea in i= Q i at most n n times. It follows that the minimum numbe of colos in {,..., } that we need to colo the emaining vetices in i= Q i is n n i= in i n n ; the total numbe of vetices in i= Q i that must be coloed with a colo in {,..., } divided by how many times each colo can be used. Given that this expession simplifies to + i= in i n n this value will neve be geate than. Note that by ou assumption, j= jn j < Since j= jn j cannot be negative, this implies liewise, n n is stictly geate than 0. { n n i= in i n n and i= in i is neve positive, ( ) n n. ( ) n n is stictly geate than 0 and Let us distibute the colos in,..., such that each colo appeas n n times in i= Q i. We will distibute them such that we place one vetex of colo in each consecutive Q i beginning with Q. When all n n vetices of a colo ae placed, the colo will be inceased by and placed in the next Q i, etuning to Q afte a colo has been placed in Q. Afte placing these colos, thee ae n n i n i i= n n i= in i n n } ( n n ) 5
Chomatic Villainy vetices uncoloed in i= Q i that still need to be coloed with a colo in {,,..., }. Note that n The value of It holds that n = n = n i n i i= n n i= n n i= in i n n i n i + i= in i n n n i n i n + i= i= ( in i n n n = i n i i= in i i= n n = i= in i n n = + i= in i n n i= in i n n + i= in i n n ( n ( n n ( n is stictly geate than ( n n ) i= n n ) ( + n n n ) ( + n ) ) i= in i n n i= i= i n i > + = i= = 0, i n i i n i. ( n ( n ) n ) n n ) ) and less than o equal to i= i n i i= in i n n i n i i= i n i ( n i= in i n. n n ) 6
Chomatic Villainy and + i= in i n n ( n n ) i= i n i + = n = n n n i= in i n n i= i n i +. i= i n i ( n i= i n i n ) Theefoe, the numbe of vetices that still need to be coloed with a colo in {,,..., } is stictly geate than 0 and less than o equal to n n. Thus, these vetices can be coloed with one colo. These vetices eceive colo the fist vetex to eceive colo of colo n n i= in i n n n n i= in i n n n n i= in i n n. The est of the vetices of colo. We will place this colo such that is placed in the next Q i afte the last vetex n n i= in i n n will be placed in consecutive Q i s such that the colo will be placed in Q afte a colo has been placed in Q. The emaining vetices in i= Q i will be coloed with a colo in { +, +,..., χ(g)}. Note that n n is equivalent to n (mod ). Theefoe, n n <. Since each colo in {,..., } appeas at most n (mod ) times in i= Q i and these colos ae distibuted acoss patite sets as evenly as possible, the colos in this set ae distibuted in i= Q i such that no colo appeas moe than once in a given Q i. Theefoe, each P i has at most n + vetices of each colo in {,..., }. n Note that fo all i and j whee i, j {,..., }, patite set P i contains exactly o + vetices with colo j. Moeove, at most colos in {,... } appea n + n n i= in i n n times in some patite set P j. The vetices in patite sets {P +,..., P χ(g) } ae coloed with colos in the set {,..., }. Theefoe, in econstucting a pope coloing of G, thee ae at most n 7
Chomatic Villainy n + n n i= in i n vetices that do not need to be ecoloed, and n B(G) j n j j= n We believe B(G) j= jn j n n n j= jn j n n. n n j= jn j n n shown. A less stict uppe bound of B(G) j= jn j n as well, but this has yet to be n n j= jn j poven. Note that since n n is equivalent to n (mod ), it holds that n stictly less than. Thus, j n j j= n has been n is n n j= jn j n n n j n j j= n n j= jn j. Theoem. Let G = Kn,..., n }{{},n,..., n,...,n }{{},..., n be a complete multipatite gaph with j patite }{{} sets of size n j fo all j. Note that n n... n > 0. If we have n > j n j and j= ( j n j < n j= n ), then Poof. Fo j B(G) j n j j= n n n j= jn j. [ ] j= j, let P j be a patite set in G. Let c be a pope coloing of V(G) that uses χ(g) colos. Theefoe all vetices in each patite set get the same colo unde c and χ(g) = i= colos ae used. We define c such that P, P,..., P ae the sets of size n and each vetex in P i whee i [ ] has colo i. Note that j= jn j is the total numbe of vetices in χ(g) i= + P i. Let c be the eaangement of c that has the highest villainy. It holds that unde any ecoloing, at least n i= in i vetices in i= P i must have a colo in {,,..., }. Any vetex in 8
Chomatic Villainy i= P i with a colo in { +, +,..., χ(g)} is coloed incoectly. Liewise, any vetex in χ(g) j= + P j with a colo in {,,..., } is coloed incoectly. Thus, in c, as many vetices as possible in i= P i ae ecoloed with a colo in { +, +,..., χ(g)}. It follows that exactly n i= in i vetices in i= i n i ae coloed with a colo in {,,..., } unde c. Let us epesent the coloing unde c of i= P i as a weighted bipatite gaph with patite sets A and B whee the vetices in A epesent the set of colos in {,,..., } and the vetices in B epesent the patite sets in {P,..., P }. An edge between a vetex f in A and g in B with weight indicates that thee ae vetices with colo f in P g. Let the weight of such an edge be denoted w f,g. The sum of these weights is equivalent to n j= jn j, the total numbe of vetices in i= P i with colos in {,,..., }. Conside the maximum weighted matching between A and B. Note that the bipatite gaph is complete, thus the maximum weighted matching is also a pefect matching. By choosing to ecolo each patite set such that the vetices in P i eceive the colo in A that was matched with i in B, we choose the eaangement of c that achieves the pope coloing of i= P i that equies the fewest numbe of ecoloings. Without loss of geneality, let us assume that in each patite set P i, the vetices of colo i do not need to be ecoloed when econstucting the pope coloing of G that equies the minimum numbe of ecoloings. Theefoe, the sum of the edges in the maximum weighted matching is w, + w, +... + w,. Fo each fixed intege s, with s {,..., }, if we ecolo the vetices in i= P i in such a way that all vetices in P i get final colo i + s modulo (note that if i + s 0 (mod ), the vetices in P i will eceive colo ), then each element w i,j whee i, j {,..., } will be included in a sum exactly once in the set {w, + w, +... + w,, w, + w, +... + w,,..., w, + w, +... + w, }. The sum of the elements in this set can be expessed as i= j= w i,j and will be the sum of diffeent values of w i,j. Note that this value epesents the sum of the weights of all edges in the weighted bipatite gaph between A and B. Theefoe, this sum is the total numbe of vetices with colos {,..., } in i= P i. Let us assume that an impope coloing of G exists such that the numbe of vetices that do not n need to be ecoloed is less than n n + j= jn j. This implies that the maximum weighted matching between A and B is less than n +. If the sum of n n j= jn j the weights in the maximum weighted matching, w, + w, +... + w,, is less than n + 9
Chomatic Villainy n n j= jn j, then each element in the set {w, + w, +... + w,, w, + w, +... + w,,..., w, + w, +... + w, } must also be less than n + that thee ae elements in this set, the maximum possible value of ) by ( n + n n j= jn j x + y = x + y. It follows that: ( n + The value of Theefoe, n n j= jn j i=. Given j= w i,j is given. Note that if x o y is an intege, it holds that n ) ( n j= jn j ) = n + n n + j= jn j j= jn j is stictly less than j= jn j n ( = n + n + ) = (n + j= j n j = n + j= j n j. ) j= j n j + and geate than o equal to j= jn j. ( ) j= j n j j= j n j n + < n + + = n = n j= j= j n j + j n j and i= j= w i,j is stictly less than n j= jn j. This is a contadiction given that the sum of weighted edges of the multipatite gaph between A and B must be equal to n j= jn j. Theefoe, the numbe of vetices that do not need to be ecoloed is at least n + n n j= jn j. Note that all vetices in χ(g) + P i ae coloed with a colo in the set {,,..., } and ae thus coloed incoectly. Theefoe, B(G) j n j j= n n n j= jn j. 0
Chomatic Villainy Conside case whee n j= jn j and j= jn j ( ) n n. Let G = Kn,..., n }{{},n,..., n,...,n }{{},..., n }{{}. Let G be popely coloed such that P i eceives colo i and {P, P,..., P } ae the pats of size n. We aim to impopely ecolo i= P i such that the colos in {,,..., } ae distibuted as evenly as possible. We fist want to distibute as many of the n i= in i vetices with a colo in {,,..., } such that each colo is in a patite set the same numbe of times and thee is the same numbe of vetices with each colo in each patite set. Thus, each colo appeas in each patite set times. This leaves n i= in i vetices in i= P i left to be n i= in i n i= in i ecoloed with a colo in {,,..., }. Let Q i be the uncoloed vetices in P i fo i [ ]. We aim to ecolo these vetices in i= Q i such that at most one vetex of each colo in {,,..., } is in a given Q i and the colos ae distibuted acoss as few patite sets as possible. The minimum numbe of patite sets these colos can be distibuted ove is given by n n i= in i i= in i. Howeve, patite set n n i= i n i i= in i {,,..., }. We colo uncoloed vetices in the fist sets with each colo in {,,..., }. This leaves n n i= in i i n i i= may not contain additional colos in n n n i= i n i i= in i n i= in i patite i= in i
Chomatic Villainy vetices that need to be ecoloed with a colo in {,,..., }. We colo n n i= in i vetices in patite set whee i n n i= in i i n i i= n n i= i n i i= in i i n i i= n n i= in i i= in i such that thee is a vetex of each colo i n n i= in i i= in i. n n i= i n i The fist i= in i patite sets will contain n i= in i + of a distinct colo while the est contain at most n i= in i vetices of the same colo. If we ecolo the sets with a colo that appeas + n n i= i n i i= in i vetices in the fist times and the est of the patite sets with a colo that appeas n i= in i n n i= in i + i= in i = n i= in i + n = n + = i= i n i n i= in i n i= in i n i= in i n i= in i times, at most + i= i n i vetices in G will not need to be ecoloed.
Chomatic Villainy Conside G = K 8,8,8,0. A pope coloing of K 8,8,8,0 is given in Figue 6. In this case, ( n n ) ( = 8 while i= in i = 0 and it holds that i= in i ) 8 = 6 ( ) n n. Let us define P i as the patite set that contains vetices with colo i unde the pope coloing given in Figue 6. We fist ecolo 9 vetices in each patite set of size 8 with each colo in {,, } as shown in Figue 7. Thee ae five vetices left in i= P i that need to be ecoloed with a colo in {,, } and these vetices will be in P and P. We will colo an additional vetex in P with each colo in {,, } and one vetex in P with each colo in {, }. The emaining vetices in i= P i eceive colo. This ecoloing is given in Figue 8. In the coloing given in Figue 8, thee ae at most two vetices in a patite set that appea in G eight times and ae the same colo. Howeve, only P and P contain a colo that appeas two times. Recoloing the vetices in P with colo and the vetices in P with colo equies ecoloings. Recoloing P with colo equies 7 ecoloings. Thus, to etun the gaph in Figue 8 to a pope coloing, 9 vetices need to be ecoloed and 5 vetices ae aleady coectly coloed. Note that n i= n 8() 0 = = 5. Figue 6: A pope coloing of K 8,8,8,0
Chomatic Villainy Figue 7: A patial ecoloing of K 8,8,8,0 Figue 8: An impope ecoloing of K 8,8,8,0 Theoem. Let G = Kn,..., n }{{},n,..., n,...,n }{{},..., n be a complete multipatite gaph with j patite }{{} sets of size n j fo all j [], with n n... n > 0. If we have n > j n j and j= ( j n j n j= n ), then Poof. Fo l B(G) = j n j j= n j= jn j. [ ] j= j, let P l be a patite set in G. Let c be a pope coloing of G that uses χ(g) colos; we define c such that P, P,..., P ae the sets of size n and each vetex in P l has colo l unde c. Note that χ(g) = j= j.
Chomatic Villainy Let c be the eaangement of c that has the highest villainy. It holds that unde any ecoloing, at least n i= in i vetices in i= P i must have a colo in {,,..., }. Any vetex in i= P i with a colo in { +, +,..., χ(g)} is coloed incoectly. Liewise, any vetex in χ(g) j= + P j with a colo in {,,..., } is coloed incoectly. Thus, in c, as many vetices as possible in i= P i ae ecoloed with a colo in { +, +,..., χ(g)}. It follows that exactly n i= in i vetices in i= i n i ae coloed with a colo in {,,..., } unde c. We need to ecolo the vetices of G in such a way that a pope coloing of G using the same set of colos is obtained. Fo integes a and b with a, b {,..., }, let f a,b be the total numbe of vetices in P a having colo b. Thus, a= b= f a,b = n j= jn j. Note that fo the vetices in patite sets P, P,..., P to be popely coloed, the vetices in each patite set must be coloed with a unique colo in the set {,,..., }. Let c s be a pope coloing of G whee the vetices in P i eceive colo i + s (mod ) fo all i [] whee s {0,,..., }. If i + s (mod ) is 0, the vetices in patite set i eceive colo. Note that thee ae coloings in the set {c 0, c,..., c }. Let us define F s as the numbe of vetices in c that do not need to be ecoloed to achieve coloing c s. It holds that F s = f,+s ( mod ) + f,+s ( mod ) +... + f, +s ( mod ) fo a given s. In c, it holds that n j= j n j = ( f, + f, +... + f, ) }{{} + ( f, + f, +... + f, ) }{{} P +... + ( f, + f, +... + f, ) }{{} P = ( f, + f, +... + f, ) + ( f, + f, +... + f,) +... + ( f, + f, +... + f, ) = F 0 + F +... + F. P Theefoe, max (F s) 0 s n j= jn j. 5
Chomatic Villainy Thus, the uppe bound on the smallest numbe of ecoloings needed to popely ecolo the vetices in i= P n i is given by n j= jn j. Since thee ae j= jn j vetices in χ(g) i= + P i with a colo in the set {,,..., }, we have n j= B(G) n jn j + = i n i i= n j= jn j. j= j n j Let c be a pope χ(g)-coloing of G. To pove B(G) i= n in i j= jn j, we define a coloing c on G that is a eaangement of c. Fo each a {,,..., } and b {,,..., } n j= we colo jn j vetices in P a with colo b. We will ecolo an additional j= j n j vetices in i= P i with colos in { +, +,..., χ(g)}. This esults in j= P j having n n j= jn j j= jn j vetices that ae not yet coloed. These vetices must be coloed with a colo fom the set {,,..., }. We aim to colo these emaining vetices in i= P i such that they ae placed in the fewest possible numbe of patite sets and thee ae no moe n than j= j n j of a colo in a given patite set. It holds that the fewest numbe of patite n sets the emaining n j= jn j j= jn j uncoloed vetices can be placed in is n j= n j n j j= jn j patite set P j whee j,,..., set {,,..., }. This leaves n n j= jn j. Let us colo a vetex that has not yet been ecoloed in each n n j= jn j j= jn j j n j j= n n j= jn j j= jn j with each colo in the 6
Chomatic Villainy vetices uncoloed in j= P j. Note that n n j= jn j = n = n j n j j= n j= jn j n j= jn j j n j j= j= j= j n j + j= j n j = j= j n j = + j= j= j n j j n j. n n j= jn j j= jn j j n j n + n n j= jn j j= jn j n j= jn j + The value of holds that j= j n j is less than o equal to j= jn j and stictly geate than j= jn j. It j= j n j + j= ( ) j= j n j j n j + =, j= j n j and j= j n j + j= j n j > + = + = 0. ( ) j= j n j j= j n j j= j n j j= j n j Theefoe, the numbe of uncoloed vetices in j= P j that need to eceive a colo in {,,..., } 7
Chomatic Villainy is in {,,..., }. We then colo + j= j n j j= jn j vetices in patite set n n j= jn j j= jn j with each colo in the set { j= j n j,,..., + j= j n j }. It holds that fo all a and b whee a, b {,,..., }, patite set P a contains exactly n j= jn j o n j= jn j + vetices with colo b. Moeove, n n j= jn j patite sets contain a colo in {,,..., } that appeas j= jn j n j= jn j + times. The vetices in patite sets {P +,..., P χ(g) } ae coloed with colos in the set {,,..., }. It follows that, in econstucting a pope coloing of G, the numbe of vetices that do not need to be ecoloed is n j= jn j + = n n j= jn j j= jn j n j= jn j = n + = j= j n j n j= jn j + n. n j= jn j + j= j n j Theefoe, B(G) j n j j= n j= jn j. 8
Chomatic Villainy Let G = K n,...,n be a complete -patite gaph with patite sets of size n. In this case, i= in i = 0 and i= in i will always be less than o equal to n n. When i= in i < n n, G is a subcase of case and will be incoectly coloed using the same method. It follows that the villainy of G is geate than o equal to j= jn j n o equal to n n n n n j= jn j n n and less than n n. Howeve, given that i= in i = 0, it holds that n n n n n n = n and n n n n = n n. Conside K 8,8,8. Let c be a pope -coloing of K 8,8,8 as shown in Figue 9. Figue 9: A pope coloing of K 8,8,8 Figue 0: An impope ecoloing of K 8,8,8 9
Chomatic Villainy Let us define P i as the patite set that eceives colo i unde c. Following the ecoloing pocedue given fo case, K 8,8,8 is impopely coloed as shown in Figue 0. Evey patite set contains thee vetices with a distinct colo. If we ecolo the vetices in P with, the vetices in P with and the vetices in P with, 5 ecoloings will be pefomed to achieve a pope coloing. Note that n 8 n = (8) = 5 and n n = (8) 8 = 6. Theoem 5. Let G = K n,...,n be a -patite gaph with patite sets of size n whee does not divide n. It holds that n n n B(G) n. Poof. Let G = K n,...,n and let P, P,...,P be the patite sets of G. By ou assumption, does not divide n. Given that each patite set must eceive a unique colo, χ(g) =. Let c be a pope -coloing of G in which the vetices in P i eceive colo i and let c be a eaangement of c. By Theoem, less than o equal to j= jn j n n n j= jn j vetices in c need to be ecoloed to estoe G to a pope coloing. Howeve, given that i= in i = 0, it holds that n n n n n = n = n = n n n n n n n + and B(G) n n. Accoding to Theoem, it holds that B(G) n n n n n. Howeve, given n 0
Chomatic Villainy that i= in i = 0, it holds that n n n n n n n = n n = n. n n n n It holds that n in case. Theefoe, n + = n. It follows that: n n ( n = n = n n. ) + Thus, n B(G) n. When i= in i = n n = 0, G is a subcase of case and will be incoectly coloed using the same method. Thus, G has a villainy of j= n jn j j= jn j. Howeve, given that i= in i = 0, this can be simplified to j n j j= n = n = n n. n Given that n in this case, n n n = n. Conside G = K 8,8,8,8. Let c be a pope -coloing of K 8,8,8,8 as shown in Figue. Let us define P i as the patite set that eceives colo i unde c. Following the ecoloing pocedue given fo case, K 8,8,8,8 is impopely coloed as shown in Figue. Evey patite set contains two vetices with a distinct colo. If we ecolo the vetices in P with, the vetices in P with the vetices in P with, and the vetices in P with, ecoloings will be pefomed to acheive
Chomatic Villainy Figue : A pope coloing of K 8,8,8,8 Figue : An impope coloing of K 8,8,8,8 a pope coloing. Note that n 8 n = (8) =. Theoem 6. Let G = K n,...,n be a -patite gaph with patite sets of size n whee divides n. It holds that n B(G) = n. Poof. Let G = K n,...,n and let P, P,...,P be the patite sets of G. Given that each patite set must eceive a unique colo, χ(g) =. Let c be a pope -coloing of G in which the vetices in P i
Chomatic Villainy eceive colo i and let c be a eaangement of c. Ou aim is to ecolo at most n n vetices of G in c in ode to obtain a pope coloing. Let us epesent the coloing unde c of i= P i as a weighted bipatite gaph with patite sets A and B whee the vetices in A epesent the set of colos in {,,..., } and the vetices in B epesent the patite sets in {P,..., P }. An edge between a vetex f in A and g in B with weight h indicates that thee ae h vetices with colo f in P g. Let the weight of such an edge be denoted w f,g. The sum of these weights is equivalent to n, the total numbe of vetices in i= P i. Conside the maximum weighted matching between A and B. Note that the bipatite gaph is complete, thus the maximum weighted matching is also a pefect matching. Without loss of geneality, let us assume the maximum weighted matching is given by w, + w, +... + w, By choosing to ecolo each patite set such that the vetices in P i eceive the colo in A that was matched with i in B, we choose the eaangement of c that achieves the pope coloing of i= P i that equies the fewest numbe of ecoloings. Let us assume that an impope coloing of G exists such that the numbe of vetices that do not need to be ecoloed is less than n. This implies that the maximum weighted matching between A and B is less than n. If the sum of the weights in the maximum weighted matching, w, + w, +... + w,, is less than n, then each element in the set {w, + w, +... + w,, w, + w, +... + w,,..., w, + w, +... + w, } must also be less than n. Given that thee ae elements in this set, the maximum possible value of i= j= w i,j is given by n which is stictly less than n. This is a contadiction given that the sum of weighted edges of the multipatite gaph between A and B must be equal to n. Theefoe, the numbe of vetices that do not need to be ecoloed is at least n. Theefoe,B(G) n n. Given that divides n, it holds that n B(G) n. To pove that B(G) n n, we define a coloing c on the vetices of G whee c is a eaangement of c. Fo each i [n] and j [], we colo n vetices in Pi with colo j. This esults in each P i having n n vetices that ae not yet coloed. In the case whee divides n, n = n and each Pi has n ( ) n = 0 vetices that ae uncoloed. Fo each i [n] and j [], each colo j appeas in P i exactly n times and at most n vetices
Chomatic Villainy ae coloed coectly. In the case whee, does not divide n, it holds that n = n. Let us efe to the uncoloed vetices in P i as Q i. We aim to ecolo i= Q i such that each colo appeas at most once in a given Q i. So fa each colo has been used n times and can theefoe be used at most n n times in i= Q i. Note that n n is equivalent to n (mod ) and can neve be geate than. Theefoe, the colos can be distibuted ove i= Q i such that thee is at most one vetex of each colo in a given Q i. Theefoe, each patite set has at least one colo that appeas n times. Given that max i f i,j is n in each patite set, n B(G) n. We believe that the villainy of K n,...,n is n n fo all values of n and whee > 0, but a stict uppe bound has not yet been found in the case whee does not divide n. Cla et al. showed that in a uniquely coloable gaph G in which evey colo appeas the same numbe of times in its χ(g)-coloing, the villainy is equivalent to the wea villainy []. Theefoe, it also holds that B w (K n,...,n ) = B(K n,...,n ). In [], Cla et al. gave esults on the chomatic villainy of connected bipatite gaphs and the chomatic villainy of complete multipatite gaphs in which evey patite set is a diffeent size. Let G = K x,y. Without loss of geneality, in a complete bipatite gaph with two patite sets X and Y whee X = x and Y = y, eithe x = y o x > y. Accoding to Poposition. in [], if x = y, then B(G) = if x > y, then B(G) = y. x+y and If x = y, then j= jn j = 0 and the villainy of G is eithe bounded by Theoem and Theoem ( ) o is given by Theoem. When n n > 0, G falls unde Theoems and. Note that since n n > 0, does not divide n. Given that =, it holds that n must be odd.
Chomatic Villainy Without loss of geneality, let n = l +. Thus, n n l + = l + ( ) = l + l = l + l (0) =, and n n =. It follows that n n n B(G) n n n n n n = n n n n = n ( n ) = n ( ( n )) = n + ( n ) = n n = n =, 5
Chomatic Villainy and B(G) n = n n n n n n n n n = n n = n ( n ) = n n = n = ( ) n = + n = +. n Theefoe, Theoems and give esults that do not contadict those given in Poposition. in ( ) []. If x = y and n n = 0, the villainy of G is given by Theoem. Theefoe, B(G) = n = n ( ) n = n = n and Theoem confims the esult given in Poposition. in []. Note that the villainy of G is also given by Theoems 5 and 6 when x = y. When does not divide n, G falls unde Theoem 5. 6
Chomatic Villainy In this case, =. It follows that n B(G) n n = n ( n ) = n n = = n, and B(G) n n = n n = n. Theefoe, Theoem 5 gives esults that do not contadict those given in Poposition. in []. When divides n, G falls unde Theoem 6. In this case, = and n B(G) = n n = n ( n ) = n n = = n. Theefoe, Theoem 6 confims the esults given in Poposition. in []. If x > y, then =. Thus, it holds that ( n n ) = (n n ) = 0 7
Chomatic Villainy and the villainy of G is given by Theoem. Theefoe, n n B(G) = n + n = n + n n + n = n, and Theoem confims the esult given in Poposition. in []. Let G = K n,n,...,n whee n > n >... > n. Accoding to Poposition. in [], if n n + n +... + n, then B(G) = i= n i, and B(G) = V(G) othewise. When n n + n +... + n, it holds that = and ( ) n n = (n n ) = 0. Assuming n > 0, the villainy of G is given by Theoem. Thus, the villainy of G is n i i= n j= n j = = i= n i n + n j j= n j j= and Theoem gives a esult equivalent to that given in Poposition. in []. Theoem poved that B(G) = V(G) fo all G = Kn,..., n }{{},n,..., n,...,n }{{},..., n whee i n i j =i j n j fo all }{{} i, j [] whee i = j. Theefoe, ou esults do not contadict those given in by Cla et al. in []. 8
Chomatic Villainy III. Paths A path on n vetices consists of a set of vetices notated v, v,..., v n, with the only edges being v i v i+ fo i [n ] [6]. A path consisting of the vetices v, v,..., v n can be efeed to as v, v,..., v n o P n. Paths have a chomatic numbe of ; the vetices ae coloed such that the colos altenate. A path with an even length that is popely coloed with two colos will have altenating vetices of colo and. Theefoe, in a path on vetices, vetices ae coloed, and vetices ae coloed. Thee ae two ways to popely colo the path with this set of colos. In one coloing, the vetices with even indices will have colo while the vetices with odd indices have colo. In the othe, the vetices with even indices will have colo while the vetices with odd indices will have colo. It holds that, in a path on vetices, if it equies m ecoloings to achieve one pope coloing, it will equie m ecoloings to achieve the othe pope coloing. It was shown by Cla et al. in [] that in an even path on vetices, the villainy is equivalent to. Let P + = v, v,..., v +. A popely coloed odd path on + vetices will have + vetices of one colo and vetices of the second colo. Without loss of geneality, let these colos be and espectively. In ode fo P + to be popely coloed, the colos must altenate. Given that thee ae + vetices of colo and only vetices with even indices, evey vetex of an odd index must eceive colo in a pope coloing. Additionally, evey vetex with an even index must eceive colo. Theefoe, thee is only one pope -coloing of an odd path up to pemutation of the colos. In [], Cla et al. poved that the villainy of an odd path on + vetices is given by. In an odd path, an impope coloing can only be ecoloed such that the odd indices eceive colo in the case whee the numbe of each colo needs to be maintained. Howeve, if we ae consideing the wea villainy of an odd path, thee ae two coloings to compae an impope coloing to; one whee the vetices with odd indices eceive colo and one whee the vetices with odd indices eceive colo. In this case, if m vetices need to be ecoloed to etun an impope coloing to a pope coloing whee the vetices with odd indices eceive colo, it will 9