Chromaticity of Turán Graphs with At Most Three Edges Deleted

Transcription

1 Iranian Journal of Mathematical Sciences and Informatics Vol. 9, No. 2 (2014), Chromaticity of Turán Grahs with At Most Three Edges Deleted Gee-Choon Lau a, Yee-hock Peng b, Saeid Alikhani c a Faculty of Comuter and Mathematical Sciences, Universiti Teknologi MARA (UiTM) Shah Alam, Selangor, Malaysia. b Deartment of Mathematics, and Institute for Mathematical Research, Universiti Putra Malaysia, UPM Serdang, Malaysia. c Deartment of Mathematics, Yazd University, , Yazd, Iran. laugc@johor.uitm.edu.my yheng@fsas.um.edu.my alikhani@yazd.ac.ir Abstract. Let P(G,λ) be the chromatic olynomial of a grah G. A grah G is chromatically unique if for any grah H, P(H,λ) = P(G,λ) imlies H is isomorhic to G. In this aer, we determine the chromaticity of all Turán grahs with at most three edges deleted. As a by roduct, we found many families of chromatically unique grahs and chromatic equivalence classes of grahs. Keywords: Chromatic olynomial, Chromatic uniqueness, Turán grah Mathematics subject classification: 05C15, 05C60. Corresonding Author Received 01 December 2012; Acceted 07 May 2014 c 2014 Academic Center for Education, Culture and Research TMU 45

2 46 G.C. Lau, Y.H. Peng, S. Alikhani 1. Introduction All grahs considered in this aer are finite and simle. Grah olynomials are a well-develoed area useful for analyzing roerties of grahs. Chromatic olynomial, characteristic olynomial and domination olynomial are some examles of these olynomials (see [1, 2, 6]). For a grah G, we denote by P(G;λ) (or P(G)), the chromatic olynomial of G. Two grahs G and H are said to be chromatically equivalent (simly χ-equivalent), denoted G H if P(G) = P(H). Let [G] denote the equivalence class determined by the grah G. AgrahGissaidtobechromatically unique(simlyχ-unique)if[g] = {G}. A family G of grahs is said to be chromatically closed (simly χ-closed) if for any grah G G, P(H) = P(G) imlies that H G. For two families of grahs G 1 and G 2, if P(G 1 ) P(G 2 ) for each G 1 G 1 and each G 2 G 2, then G 1 and G 2 are said to be chromatically disjoint (simly χ-disjoint). Many families of χ-unique grahs are known (see [7, 8, 11, 12, 13]). For a grah G, let e(g), v(g), t(g), Q(G), K(G) and χ(g) resectively be the number of vertices, edges, triangles, induced 4-cycle subgrah, comlete subgrah K 4 and chromatic number of G. By G, we denote the comlement of G. Let O n be an edgeless grah with n vertices. Suose S is a set of s edges of G. Denote by G S the grah obtained from G by deleting all edges in S and by S the grah induced by S. For t 2 and t, let F = K( 1, 2,..., t ) be a comlete t-artite grah with artition sets V i such that V i = i fori = 1,2,...,t. TheTurángrah, denotedt = K(t 1,t 2 (+1)), is the unique comlete t-artite grah having t 1 1 artite sets of size and t 2 artite sets of size +1. In this aer, we determine the chromaticity of all Turán grahs with at most s ( 3) edges deleted for s+2. As a by roduct, we found many families of chromatically unique grahs and equivalent classes of grahs. 2. Preliminary results and notations Let K s ( 1, 2,..., t ) be the family {K( 1, 2,..., t ) S S E(G) and S = s}. For 1 s + 1, we denote by K K(1,s) i,j ( 1, 2,..., t ) the grah in

3 Chromaticity of Turán grahs with at most three edges deleted 47 K s ( 1, 2,..., t ) where the s edges in S induced a K(1,s) with center in V i and all the end-vertices in V j, and by K sk2 i,j ( 1, 2,..., t ) the grah in K s ( 1, 2,..., t ) where the s edges in S induced a matching with one endvertex in V i and another end-vertex in V j. For convenience, we let T s = K s (t 1,t 2 (+1)), T K(1,s) i,j = K K(1,s) i,j (t 1,t 2 (+1)) and T sk2 i,j = K sk2 i,j (t 1,t 2 (+1)). Hence, each grah in T s has t 1 1 artite sets of size and t 2 artite sets of size +1. For a grah G and a ositive integer k, a artition {A 1,A 2,...,A k } of V(G) is called a k-indeendent artition in G if each A i is a non-emty indeendent set ofg. Letα(G,k)denotethenumberofk-indeendentartitionsinG. IfGisof order n, then P(G,λ) = n k=1 α(g,k)(λ) k where (λ) k = λ(λ 1) (λ k+1) (see [14]). Therefore, α(g,k) = α(h,k) for each k = 1,2,..., if G H. For a grah G with vertices, the olynomial σ(g,x) = k=1 α(g,k)xk is calledtheσ-olynomialofg(see[3]),andtheolynomialh(g,x) = k=1 α(g,k) x k is called the adjoint olynomial of G (see [9]). If h(g,x) = h(h,x) then we say G is adjointly equivalent to H, denoted G h H. A family G of grahs is said to be adjoint closed (simly χ h -closed) if for any grah G G, h(h,x) = h(g, x) imlies that H G. Clearly, the conditions P(G, λ) = P(H, λ), σ(g,x) = σ(h,x) and h(g,x) = h(h,x) are equivalent for any grahs G and H. For disjoint grahs G and H, G + H denotes the disjoint union of G and H, and mg the disjoint union of m coies of G; G H denotes the grah whose vertex-set is V(G) V(H) and whose edge-set is {xy x V(G) and y V(H)} E(G) E(H). Throughout this aer, all the t-artite grahs G under consideration are 2-connected with χ(g) = t. For terms used but not defined here we refer to [16]. Lemma 2.1. (see [7]) Let G and H be two grahs with H G, then v(g) = v(h), e(g) = e(h), t(g) = t(h) and χ(g) = χ(h). Moreover, α(g,k) =

4 48 G.C. Lau, Y.H. Peng, S. Alikhani α(h,k) for each k = 1,2,..., and Q(G)+2K(G) = Q(H)+2K(H). Note that if χ(g) = 3, then G H imlies that Q(G) = Q(H). Lemma 2.2. (Brenti [3]) Let G and H be two disjoint grahs. Then σ(g H,x) = σ(g,x)σ(h,x). In articular, t σ(k(n 1,n 2,...,n t ),x) = σ(o ni,x). i=1 The above lemma is equivalent to the following: Remark. Let G and H be two disjoint grahs. Then h(g+h,x) = h(g,x)h(h,x). In articular, t h(k(n 1,n 2,...,n t ),x) = h(k ni,x). i=1 For an edge e = v 1 v 2 of a grah G, then G e is defined as follows: the vertex set of G e is (V(G)\{v 1,v 2 }) {u}, and the edge set of G e is {e e E(G),e is not incident with v 1 or v 2 } {uv v N G (v 1 ) N G (v 2 )}. For examle, let e 1 be an edge of C 4 and e 2 an edge of K 4. Then C 4 e 1 = K 1 +K 2 and K 4 e 2 = K 3. Lemma 2.3. (Liu [10]) Let G be a grah with e E(G). Then h(g,x) = h(g e,x)+h(g e,x), In articular, if e = uv does not belong to any triangle of G, then h(g,x) = h(g e,x)+xh(g {u,v},x), where G e (resectively G {u,v}) denote the grah obtained by deleting the edge e (resectively the vertices u and v) from G. Denote by β(g) the minimum real root of h(g).

5 Chromaticity of Turán grahs with at most three edges deleted 49 Lemma 2.4. (Zhao [17]) Let G be a connected grah such that G contains H as a roer subgrah. Then β(g) < β(h). Lemma 2.5. (Teo and Koh [15]) The grah K(,q) is χ-unique for all q 2. Suose F = K( 1, 2,..., t ) and G = F S for a set S of s edges of G. A non-emty indeendent set of G is imroer if it is not a subset of a artite set of G. Otherwise, it is roer. A k-indeendent artition in G is imroer if it contains at least one imroer indeendent set. Define α k (G) = α(g,k) α(f,k) for k t+1. Hence, α k (G) is the number of k-indeendent artitions in G that contains at least one imroer indeendent set. Lemma 2.6. (Zhao [17]) Let F = K( 1, 2,..., t ) and G = F S. If 1 s+1, then s α t+1 (G) = α(g,t+1) α(f,t+1) 2 s 1, α t+1 (G) = s if and only if the subgrah induced by any r 2 edges in S is not a comlete multiartite grah, and α t+1 (G) = 2 s 1 if and only if all s edges in S share a common end-vertex and the other end-vertices belong to the same V i for some i. Lemma 2.7. (Dong et al. [5]) Let 1, 2 and s be ositive integers with 3 1 2, then (i) K K(1,s) 1,2 ( 1, 2 ) is χ-unique for 1 s 2 2, (ii) K K(1,s) 2,1 ( 1, 2 ) is χ-unique for 1 s 1 2, and (iii) K sk2 ( 1, 2 ) is χ-unique for 1 s 2 1. Lemma 2.8. (Zhao [17]) Let s 1, 2 and t 1 1. If s+2, then T s is χ-closed. Lemma 2.9. (Zhao [17]) Suose s 1 and s+2, then

6 50 G.C. Lau, Y.H. Peng, S. Alikhani (i) every T K(1,s) i,j is χ-unique for any (i,j) where 1 i j t and V i = V j =, or V i =, V j = +1, or V i = +1, V j =, or V i = V j = +1. (ii) T sk2 1,2 is χ-unique if t 1 = Grahs in T s for S = s 2 and s+2 It is well known that the Turán grah is χ-unique [4]. By Lemma 2.9, all grahs in T 1 are χ-unique for 3. We shall now determine the chromaticity of all grahs in T 2 for 4. Let G be a grah in T 2. Denote by G the (disjoint union of all) non-comlete comonent(s) of G. It is then easy to verify by exhaustive construction that T 2 contains 25 non-isomorhic grahs, named G 2,i (1 i 25), withthegrahg 2,i shownintable1. Notethateach circle associated with G 2,i is a comlete grah of order or +1 as indicated. Suose F = K( 1, 2,..., t ). For G = F S, denote by t i (G) the number of triangles in G that contains i deleted edges in S for i = 1,2,3. Suose G T s. An edge e = uv in S is of Tye A (resectively, Tye B and Tye C) if u V i, v V j for 1 i < j t 1 (resectively, for 1 i t 1, t j t, and for t i < j t). Denote by s 1 (G) (resectively, s 2 (G) and s 3 (G)) the number of Tye A (resectively, Tye B and Tye C) edges in S. By Lemma 2.6, we have 2 α t+1 (G) 3. Note that for 1 i 25, t(g 2,i ) = t(f) 2(t 1 +t 2 (+1))+4+k where0 k = s 2 (G 2,i )+2s 3 (G 2,i )+t 2 (G 2,i ) 5. We comute the ordered air (α t+1 (G 2,i ),k) for each G 2,i (1 i 25), and the results are shown in Table 1. We then artition the family T 2 according to the value (α t+1 (G 2,i ),k), and each art of the artition is denoted by G(α t+1 (G 2,i ),k). Hence, we have the following classification of the grahs.

7 Chromaticity of Turán grahs with at most three edges deleted 51 Table 1 (1 of 2): Non-comlete comonent(s) of G 2,i (1 i 25) for G 2,i K 2 (t 1,t 2 (+1)) G 2,1 G 2,2 G 2,3 G 2,4 G 2,5 (2,0) (2,2) (2,4) G 2,6 G 2,7 G 2,8 (3,0) +1 (3,2) (3,2) (3,4) (2,0) G 2,9 G 2,10 G 2,11 G 2, (2,1) +1 (2,2) +1 (2,2) (2,3) +1

8 52 G.C. Lau, Y.H. Peng, S. Alikhani Table 1 (2 of 2): Non-comlete comonent(s) of G 2,i (1 i 25) for G 2,i K 2 (t 1,t 2 (+1)) G 2,13 G 2,14 G 2,15 G 2, (2,4) G 2, (2,1) (2,2) (2,3) G 2,18 G 2,19 G 2, (2,3) (2,4) (2,5) (2,0) G 2,21 G 2,22 G 2,23 G 2, (2,1) (2,2) (2,2) (2,3) +1 G 2, (2,4) +1 G 1 = G(2,0) = {G 2,1 ;G 2,8 ;G 2,20 }; G 2 = G(2,1) = {G 2,9 ;G 2,14 ;G 2,21 }; G 3 = G(2,2) = {G 2,2 ;G 2,10 ;G 2,11 ;G 2,15 ;G 2,22 ;G 2,23 }; G 4 = G(2,3) = {G 2,12 ;G 2,16 ;G 2,17 ;G 2,24 }; G 5 = G(2,4) = {G 2,3 ;G 2,13 ;G 2,18 ;G 2,25 }; G 6 = G(2,5) = {G 2,19 }; G 7 = G(3,0) = {G 2,4 }; G 8 = G(3,2) = {G 2,5 ;G 2,6 }; G 9 = G(3,4) = {G 2,7 }. Note that for an edgeless grah O n, α(o n,2) = 2 n 1 1 and α(o n,3) = 1 3! (3n 3 2 n +3). We now resent our main theorem of this section.

9 Chromaticity of Turán grahs with at most three edges deleted 53 Theorem 3.1. For integers 4, t 1 1 and t 1 + t 2 3, all the grahs in T 2 are χ-unique excet that {G 2,16,G 2,17 t 1,t 2 2} is a χ-equivalence class. Proof. By Lemma 2.8, T 2 is χ-closed. Observe that for 1 i < j 9 and each grah H G i and each grah H G j, either α t+1 (H ) α t+1 (H ) or t(h ) t(h ). By Lemma 2.1, H H. Hence, each G i and G j (1 i < j 9) are χ-disjoint. Since T 2 is χ-closed, we conclude that each G i (1 i 9) is χ-closed. It follows immediately that G 2,4,G 2,7 and G 2,19 are χ-unique. By Lemma 2.9(i), we also know that G 2,5 and G 2,6 are χ-unique. We now determine the chromaticity of the grahs in G i for 1 i 5. It suffices to show that for any two grahs G and G in G i, either α k (G ) α k (G ) for k = t + 2 or t + 3, or β(g ) β(g ). Otherwise, we shall show that h(g,x) = h(g,x). Since the roofs in determining the chromaticity of each grah in G i are similar for 1 i 5, we shall only elaborate in more details for grahs in G 2 and G 3. (1). Grahs in G 1. We first note that by Lemma 2.9(ii), G 2,1 is χ-unique. Now observe that if G 2,8 G 2,20, then Lemma 2.2 imlies that h(k + G 2,8,x) = h(g 2,20,x). However, Lemma 2.4 imlies that β(k + G 2,8) = β(g 2,8) < β(g 2,20), a contradiction. Hence, all the grahs in G 1 are χ-unique. (2). Grahs in G 2. We comute α t+2 (G) for each G G 2. Let V = {V 1,V 2,...,V t } be the unique t-indeendent artition of G T 2. Suose V = {V 1,V 2,...,V t,v t+1,v t+2} is an imroer (t + 2)-indeendent artition in G. Since 4, V has at most two imroer indeendent sets. If V contains exactly two imroer indeendent sets, say V t+1 and V t+2, then V i \(V t+1 V t+2) is a roer indeendent set in V for each 1 i t. If V contains exactly one imroer indeendent set, say V t+2, then we may assume that V i V t+2 = for i = 1,2,...,t 2, and V i V t+2 for i = t 1,t. Hence, there exist exactly two roer indeendent sets in V, say V t and V t+1

10 54 G.C. Lau, Y.H. Peng, S. Alikhani such that V t V t+1 = V for V {V 1,V 2,...,V t 2,V t 1 \V t+2,v t \V t+2}. Define f 1 = (t 1 +2)(2 1 1)+(t 2 2)(2 1), f 2 = (2 2 1)+t 1 (2 1 1)+(t 2 1)(2 1), f 3 = 2(2 2 1)+(t 1 2)(2 1 1)+t 2 (2 1). Hence, α t+2 (G 2,i ) = 1+f 2 +f 3 for i = 9,21, α t+2 (G 2,14 ) = 2f 3. Since 4, α t+2 (G 2,14 ) α t+2 (G 2,i ) = > 0fori = 9,21. Wenowshow that G 2,9 G 2,21. By Lemma 2.4, β(g 2,9 ) = β(g 2,9) < β(g 2,21) = β(g 2,21 ), a contradiction. Hence, all the grahs in G 2 are χ-unique. (3). Grahs in G 3. By a similar argument as in (2) above, we have for each grah G G 3, α t+2 (G 2,i ) = 1+2f 2 for i = 2,10,11,23, α t+2 (G 2,15 ) = f 2 +f 3, α t+2 (G 2,22 ) = 1+f 1 +f 3. Since 4, α t+2 (G 2,15 ) α t+2 (G 2,i ) = > 0 for i = 2, 10, 11, 22, 23. We need to comute α t+3 (G 2,i ) for i = 2, 10, 11, 22, 23. Suose V = {V 1,V 2,...,V t,v t+1,v t+2,v t+3} is an imroer (t + 3)-indeendent artition in G. If V contains exactly two imroer indeendent sets, say V t+2 and V t+3, then exactly one of V i \(V t+2 V t+3) (1 i t) needs to be artitioned into two roer indeendent sets. Note that V has at most two imroer indeendent sets. If V contains exactly one imroer indeendent set, say V t+3, then either exactly one of V i \V t+3 (1 i t) needs to be artitioned into three roer indeendent sets, or else exactly two of V i \V t+3 (1 i t)

11 Chromaticity of Turán grahs with at most three edges deleted 55 need to be artitioned into two roer indeendent sets resectively. Define [ ] t 1 +2 g 1 = ( )+ t 2 2 ( ) + 3! 3! Hence, g 2 = g 3 = [ (t1 ) +2 (2 1 1) 2 +(t 1 +2)(t 2 2)(2 1 1)(2 1)+ 2 ( ] t2 2 )(2 1) 2, 2 [ 1 3! ( )+ t 1 3! ( )+ ] [ t 2 1 ( ) + t 1 (2 2 1)(2 1 1)+ 3! ( ) (t 2 1)(2 2 1)(2 t1 1)+ (2 1 1) ( ] t 1 (t 2 1)(2 1 1)(2 t2 1 1)+ )(2 1) 2, 2 [ 2 3! ( )+ t 1 2 ( )+ 3! t 2 3! ( ) ] + [ (2 2 1) 2 + 2(t 1 2)(2 2 1)(2 1 1)+2t 2 (2 2 1)(2 1)+ ( ) t1 2 (2 1 1) 2 +(t 1 2)t 2 (2 1 1)(2 1)+ 2 ( ] t2 )(2 1) 2. 2 α t+3 (G 2,2 ) = [(2 3 1)+(2 2 1)+(t 1 1)(2 1 1)+ (t 2 1)(2 1)]+2g 2, α t+3 (G 2,10 ) = [3(2 2 1)+(t 1 2)(2 1 1)+(t 2 1)(2 1)]+ 2g 2,

12 56 G.C. Lau, Y.H. Peng, S. Alikhani a t+3 (G 2,11 ) = [(2 3 1)+(t 1 +1)(2 1 1)+(t 2 2)(2 1)]+2g 2, α t+3 (G 2,22 ) = [2(2 2 1)+t 1 (2 1 1)+(t 2 2)(2 1)]+ g 1 +g 3, α t+3 (G 2,23 ) = 2(2 2 1)+t 1 (2 1 1)+(t 2 2)(2 1)+2g 2. Since 4, by Software Male, α t+3 (G 2,22 ) α t+3 (G 2,2 ) = (2 3 ) > 0, α t+3 (G 2,2 ) α t+3 (G 2,10 ) = 2 3 > 0, α t+3 (G 2,10 ) α t+3 (G 2,11 ) = 2 3 > 0, α t+3 (G 2,11 ) α t+3 (G 2,23 ) = 2 3 > 0. Hence, all the grahs in G 3 are χ-unique. (4). Grahs in G 4. For each grah G G 4, α t+2 (G 2,i ) = 1+f 1 +f 2 for i = 12,24, α t+2 (G 2,j ) = 2f 2 for j = 16,17. Since 4, α t+2 (G 2,j ) α t+2 (G 2,i ) = > 0. ByLemma2.4, β(g 2,12 ) = β(g 2,12) < β(g 2,24) = β(g 2,24 ). By Lemmas 2.2 and 2.3, we have h(g 2,16,x) = [h(k,x)] t1 2 [h(k +1,x)] t2 1 h(g 2,16,x) = [h(k,x)] t1 2 [h(k +1,x)] t2 1 [(h(k,x)) 2 h(k +1,x)+ 2xh(K 1,x)(h(K,x)) 2 ] and h(g 2,17,x) = [h(k,x)] t1 1 [h(k +1,x)] t2 2 h(g 2,17,x) = [h(k,x)] t1 1 [h(k +1,x)] t2 2 [h(k,x)(h(k +1,x)) 2 + 2xh(K 1,x)h(K,x)h(K +1,x)]. Clearly, G 2,16 G 2,17 for t 1,t 2 2. Hence, all the grahs in G 4 are χ-unique excet that {G 2,16,G 2,17 t 1,t 2 2} is a χ-equivalence class. (5). Grahs in G 5. For each G G 5, α t+2 (G 2,i ) = 1+2f 1 for i = 3,13,25,

13 Chromaticity of Turán grahs with at most three edges deleted 57 α t+2 (G 2,18 ) = f 1 +f 2. Since 4, α t+2 (G 2,18 ) > α t+2 (G 2,i ) = > 0. We now comare α t+3 (G 2,3 ), α t+3 (G 2,13 ) and α t+3 (G 2,25 ). We have α t+3 (G 2,3 ) = [2(2 2 1)+t 1 (2 1 1)+(t 2 2)(2 1)]+2g 1, α t+3 (G 2,13 ) = [(2 2 1)+(t 1 +2)(2 1 1)+(t 2 3)(2 1)]+2g 1, α t+3 (G 2,25 ) = [(t 1 +4)(2 1 1)+(t 2 4)(2 1)]+2g 1. Since 4, by Software Male, α t+3 (G 2,3 ) α t+3 (G 2,13 ) = α t+3 (G 2,13 ) α t+3 (G 2,25 ) = > 0. Hence, all the grahs in G 5 are χ-unique. The roof is now comlete. 4. Grahs in T 3 for 5 Let u,v and w be a vertex of the comlete grahs K,K q and K r resectively. Denote by K K q the grah obtained from K +K q by adding the edge uv. For convenience, let K + = K 1 K. Also let K K q K r be the grah obtained from K +K q +K r by adding two edges uv and vw. We now list all the 25 ossible structures induced by three edges deleted from the Turán grah which can be obtained by brute force construction (with resect to the artite sets) in Figure 1. Since each circle in Figure 1 is a artite set of size or +1, we obtain from the structures in Figure 1, 213 non-isomorhic grahs in T 3, named G 3,i (1 i 213). A table of the grahs G 3,i similar to that of Table 1 is available uon request. Note that each circle associated with G 3,i is a comlete grah of order or + 1 as indicated. For examle, the to left structure in Figure 1 will give us three non-isomorhic grahs G 3,1, G 3,2 and G 3,3 as shown in Figure 2. By Lemma 2.6, we have 3 α t+1 (G 3,i ) 7 for 1 i 213. Also note that for 1 i 213, t(g 3,i ) = t(f) 3(t 1 + t 2 ( + 1)) k where

14 58 G.C. Lau, Y.H. Peng, S. Alikhani 0 k = s 2 (G 3,i ) + 2s 3 (G 3,i ) + t 2 (G 3,i ) t 3 (G 3,i ) 9. We now comute the ordered air (α t+1 (G 3,i ),k) for each G 3,i (1 i 213). A table of G (3,i) with (α t+1 (G 3,i ),k) is also available uon request. We then artition the family T 3 according to the value (α t+1 (G 3,i ),k), and each art of the artition is denoted by G(α t+1 (G 3,i ),k). Figure 1. List of structures induced by three edges deleted from K(t 1,t 2 (+1)) G 3,1 G 3,2 G 3, Figure 2: Grahs obtained from to left structure in Figure 1. Hence, we have the following classification of the grahs. G 1 = G(3,0) = {G 3,1 ;G 3,15 ;G 3,35 ;G 3,83 ;G 3,104 ;G 3,140 ;G 3,168 ;G 3,204 };

15 Chromaticity of Turán grahs with at most three edges deleted 59 G 2 = G(3,1) = {G 3,19 ;G 3,36 ;G 3,59 ;G 3,84 ;G 3,105 ;G 3,114 ;G 3,141 ;G 3,148 ; G 3,169 ;G 3,170 ;G 3,186 ;G 3,205 }; G 3 = G(3,2) = {G 3,16 ;G 3,25 ;G 3,37 ;G 3,60 ;G 3,85 ;G 3,86 ;G 3,106 ;G 3,107 ; G 3,115 ;G 3,116 ;G 3,130 ;G 3,143 ;G 3,149 ;G 3,150 ;G 3,171 ;G 3,172 ; G 3,173 ;G 3,175 ;G 3,187 ;G 3,188 ;G 3,206 ;G 3,207 } G 4 = G(3,3) = {G 3,2 ;G 3,20 ;G 3,21 ;G 3,38 ;G 3,39 ;G 3,61 ;G 3,87 ;G 3,108 ;G 3,109 ; G 3,117 ;G 3,118 ;G 3,119 ;G 3,131 ;G 3,142 ;G 3,145 ;G 3,152 ;G 3,153 ; G 3,160 ;G 3,174 ;G 3,176 ;G 3,177 ;G 3,179 ;G 3,189 ;G 3,190 ;G 3,191 ; G 3,193 ;G 3,208 ;G 3,209 }; G 5 = G(3,4) = {G 3,17 ;G 3,26 ;G 3,27 ;G 3,40 ;G 3,62 ;G 3,63 ;G 3,88 ;G 3,89 ;G 3,110 ; G 3,111 ;G 3,120 ;G 3,121 ;G 3,122 ;G 3,123 ;G 3,132 ;G 3,133 ;G 3,144 ; G 3,151 ;G 3,156 ;G 3,161 ;G 3,178 ;G 3,180 ;G 3,181 ;G 3,182 ;G 3,192 ; G 3,194 ;G 3,195 ;G 3,197 ;G 3,210 ;G 3,211 }; G 6 = G(3,5) = {G 3,22 ;G 3,23 ;G 3,41 ;G 3,64 ;G 3,90 ;G 3,112 ;G 3,124 ;G 3,125 ;G 3,126 ; G 3,134 ;G 3,135 ;G 3,146 ;G 3,154 ;G 3,155 ;G 3,163 ;G 3,183 ;G 3,184 ; G 3,196 ;G 3,198 ;G 3,199 ;G 3,200 ;G 3,212 }; G 7 = G(3,6) = {G 3,3 ;G 3,18 ;G 3,28 ;G 3,29 ;G 3,42 ;G 3,65 ;G 3,91 ;G 3,113 ;G 3,127 ; G 3,128 ;G 3,136 ;G 3,137 ;G 3,147 ;G 3,157 ;G 3,158 ;G 3,162 ;G 3,165 ; G 3,185 ;G 3,201 ;G 3,202 ;G 3,213 }; G 8 = G(3,7) = {G 3,24 ;G 3,66 ;G 3,129 ;G 3,138 ;G 3,159 ;G 3,164 ;G 3,203 }; G 9 = G(3,8) = {G 3,30 ;G 3,139 ;G 3,166 }; G 10 = G(3,9) = {G 3,167 }; G 11 = G(4,0) = {G 3,4 ;G 3,43 ;G 3,51 ;G 3,92 }; G 12 = G(4,1) = {G 3,44 ;G 3,52 ;G 3,67 ;G 3,93 }; G 13 = G(4,2) = {G 3,31 ;G 3,45 ;G 3,53 ;G 3,68 ;G 3,75 ;G 3,94 ;G 3,95 ;G 3,96 }; G 14 = G(4,3) = {G 3,5 ;G 3,6 ;G 3,46 ;G 3,47 ;G 3,54 ;G 3,55 ;G 3,69 ;G 3,76 ;G 3,97 ;G 3,98 }; G 15 = G(4,4) = {G 3,32 ;G 3,48 ;G 3,56 ;G 3,70 ;G 3,71 ;G 3,77 ;G 3,99 ;G 3,100 ;G 3,101 }; G 16 = G(4,5) = {G 3,49 ;G 3,57 ;G 3,72 ;G 3,78 ;G 3,79 ;G 3,102 }; G 17 = G(4,6) = {G 3,7 ;G 3,33 ;G 3,50 ;G 3,58 ;G 3,73 ;G 3,80 ;G 3,103 }; G 18 = G(4,7) = {G 3,74 ;G 3,81 }; G 19 = G(4,8) = {G 3,34 ;G 3,82 };

16 60 G.C. Lau, Y.H. Peng, S. Alikhani G 20 = G(5,0) = {G 3,8 }; G 21 = G(5,3) = {G 3,9 }; G 22 = G(5,6) = {G 3,10 }; G 23 = G(7,0) = {G 3,11 }; G 24 = G(7,3) = {G 3,12 ;G 3,13 }; G 25 = G(7,6) = {G 3,14 }. Now we resent our main result of this section. Theorem 4.1. For integers 5, t 1 1 and t 1 + t 2 3, all the grahs in K 3 (t 1,t 2 (+1)) are χ-unique excet that {G 3,19,G 3,148 t 1 4,t 2 0}, {G 3,20,G 3,118 t 1 3,t 2 1}, {G 3,21,G 3,61,G 3,152 t 1,t 2 2}, {G 3,22, G 3,124,G 3,155 t 1,t 2 2}, {G 3,24,G 3,159 t 1 1,t 2 4}, {G 3,26,G 3,132 t 1 3,t 2 1}, {G 3,28,G 3,136 t 1,t 2 2}, {G 3,32,G 3,77 t 1 2,t 2 1}, {G 3,65,G 3,157 t 1 1,t 2 3}, {G 3,85,G 3,106 t 1 3,t 2 1}, {G 3,90,G 3,112 t 1 1,t 2 3}, {G 3,117,G 3,189,G 3,193 t 1 4,t 2 2}, {G 3,125,G 3,198 t 1 2,t 2 3}, {G 3,127,G 3,158 t 1 1,t 2 3}, {G 3,128,G 3,202 t 1 1,t 2 4}, {G 3,162,G 3,165 t 1,t 2 3}, {G 3,192,G 3,197 t 1,t 2 3} and {G 3,196, G 3,200 t 1 2,t 2 4} are χ-equivalence classes. Proof. By Lemma 2.8, T 3 is χ-closed. Observe that for 1 i < j 25 and each grah H G i and each grah H G j, either α t+1 (H ) α t+1 (H ) or t(h ) t(h ). By Lemma 2.1, H H. Hence, each G i and G j (1 i < j 25) are χ-disjoint. Since T 3 is χ-closed, we conclude that each G i (1 i 25) isχ-closed. ItfollowsimmediatelythatG 3,8,G 3,9,G 3,10,G 3,11,G 3,14 andg 2,167 are χ-unique. By Lemma 2.9(i), we also know that G 3,12 and G 3,13 are χ- unique. WenowdeterminethechromaticityofthegrahsinG i fori {1,2,...,19}\{10}. It suffices to show that for any two grahs G and G in G i, either α k (G ) α k (G ) for k = t+2, t+3 or t+4, or β(g ) β(g ). Otherwise, we shall show that h(g,x) = h(g,x). We use an argument similar to that in the roof of

17 Chromaticity of Turán grahs with at most three edges deleted 61 Theorem 3.1. In what follows, f i and g i (i = 1,2,3) are as defined in the roof of Theorem 3.1. (1). Grahs in G 1. Define d 1 = 2(2 3 1)+(t 1 2)(2 1 1)+t 2 (2 1), d 2 = (2 3 1)+2(2 2 1)+(t 1 3)(2 1 1)+t 2 (2 1), d 3 = 4(2 2 1)+(t 1 4)(2 1 1)+t 2 (2 1). So, α t+3 (G 3,1 ) = 1+3d 1 +3g 3, α t+3 (G 3,i ) = 1+3d 2 +3g 3 for i = 15,140, α t+3 (G 3,35 ) = 1+d 1 +2d 2 +3g 3, α t+3 (G 3,83 ) = 1+d 1 +2d 3 +3g 3, α t+3 (G 3,104 ) = 1+2d 2 +d 3 +3g 3 α t+3 (G 3,168 ) = 1+d 2 +2d 3 +3g 3, α t+3 (G 3,204 ) = 1+3d 3 +3g 3. Since 5, α t+3 (G 3,1 ) α t+3 (G 3,35 ) = 2 2 > 0, α t+3 (G 3,35 ) α t+3 (G 3,i ) = 2 3 > 0 for i = 15,140, α t+3 (G 3,i ) α t+3 (G 3,j ) = 2 3 > 0 for j = 83,104, α t+3 (G 3,j ) α t+3 (G 3,168 ) = 2 3 > 0, α t+3 (G 3,168 ) α t+3 (G 3,204 ) = 2 3 > 0. So, α t+3 (G 3,1 ) > α t+3 (G 3,35 ) > α t+3 (G 3,15 ) = α t+3 (G 3,140 ) > α t+3 (G 3,83 ) = α t+3 (G 3,104 ) > α t+3 (G 3,168 ) > α t+3 (G 3,204 ). Let I j k (G) be the number of k-indeendent artitions with j 1 imroer indeendent sets. Then, α k (G) = j 1 Ij k (G). We now comare α t+4(g 3,15 )

18 62 G.C. Lau, Y.H. Peng, S. Alikhani with α t+4 (G 3,140 ). α t+4 (G 3,15 ) = I 1 t+4(g 3,15 )+I 2 t+4(g 3,15 )+ [3(2 3 1)+(t 1 3)(2 1 1)+t 2 (2 1)], α t+4 (G 3,140 ) = I 1 t+4(g 3,140 )+I 2 t+4(g 3,140 )+ [(2 4 1)+3(2 2 1)+(t 1 4)(2 1 1)+t 2 (2 1)]. Note that I j t+4 (G 3,15) = I j t+4 (G 3,140) for j = 1,2. Since 5, α t+4 (G 3,15 ) α t+4 (G 3,140 ) = 2 4 > 0. We now show that h(g 3,83,x) h(g 3,104,x). By Lemmas 2.2 and 2.3, [ ] t1 4[ ] t2 [ h(g 3,83,x) = h(k,x) h(k +1,x) (h(k K,x)) 2 + ] xh(k K,x)h(K 1 K 1,x), h(g 3,104,x) = [ ] t1 4[ ] t2 [ h(k,x) h(k +1,x) (h(k K,x)) 2 + x(h(k K 1,x)) 2 ]. By Lemma 2.4, β(k K ) < β(k K 1 ). Therefore, β(g 3,83 ) < β(g 3,104 ). Hence, all the grahs in G 1 are χ-unique. The chromaticity of grahs in G i, i = {2,3,...,19}\{10} can be similarly determined as in (1). Since it is long and rather reetitive, the roofs are omitted. The roof is now comlete. A diagram of all the χ-equivalence classes obtained in Theorem 4.1 is resented in Figure 3 for easy reference and comarison. Remarks. Observe that grahs G 3,162 and G 3,165 can be obtained from G 2,16 and G 2,17 resectively with one more edge deleted. For t 1,t 2 s 2, let G (resectively, G ) be a grah in T s such that S is a star grah with the central vertex belongs to a artite set of size (resectively, size +1) and the end-vertices belong to different artite sets of size + 1 (resectively, size ). It is clear that G = G. By Lemmas 2.2, 2.3 and mathematical induction on s, it is easy to show that h(g,x) = h(g,x) for all 2.

19 Chromaticity of Turán grahs with at most three edges deleted 63 Conjecture 1. The family {G,G } is a χ-equivalence class. For integers m 1,m 2,m 3 0, let G and H be grahs in T s such that the grah induced by all the non-comlete comonents of G (resectively, H) is G 2,16+m 1 (K K )+m 2 (K K +1 )+m 3 (K +1 K +1 ) (resectively, G 2,17+ m 1 (K K )+m 2 (K K +1 )+m 3 (K +1 K +1 )) where m 1 +m 2 +m 3 1. G 3.19 G 3,148 G 3,20 G 3, G 3,21 G 3,61 G 3,152 G 3,22 +1 G 3,124 G 3, G 3,24 G 3,159 G 3,26 G 3, G 3,28 G 3,136 G 3,32 G 3, G 3,65 G 3,157 G 3,85 G 3, G 3,90 G 3,112 G 3,117 G 3,189 G 3, G 3,125 G 3,198 G 3,127 G 3, G 3,128 G 3,202 G 3,162 G 3, G 3,192 G 3,197 G 3, G 3, Figure 3: Structures of grahs in Chromatic Equivalence Classes listed in Theorem 2.

20 64 G.C. Lau, Y.H. Peng, S. Alikhani Clearly, G H but G = H. Observe that {G,H} is a χ-equivalence class if m 1 = m j = 0 for j = 2 or 3. Hence, we end this aer with the following roblem. Problem 1. Find all the values of m i,i = 1,2,3 such that {G,H} is a χ- equivalence class for G and H defined above. Acknowledgement. The authors would like to exress their gratitude to the referee for helful comments. References 1. S. Alikhani, On the domination olynomials of non P 4 -free grahs, Iran. J. Math. Sci. Inf., 8(2), (2013), G. D. Birkhoff, A determinant formula for the number of ways of colouring a ma, Annal. Math., 2nd Ser., 14, ( ), F. Brenti, Exansions of chromatic olynomial and log-loncavity, Trans. Amer. Math. Soc., 332, (1992), C. Y. Chao, G. A. Novacky Jr., On maximally saturated grahs, Discrete Math., 41, (1982), F. M. Dong, K. M. Koh, K. L. Teo, C. H. C. Little, M. D. Hendy, Shar bounds for the number of 3-indeendent artitions and the chromaticity of biartite grahs, J. Grah Theory, 37, (2001), G. H. Fath-Tabar, A. R. Ashrafi, The Hyer-Wiener Polynomial of Grahs, Iran. J. Math. Sci. Inf., 6 (2), (2011), K. M. Koh, K. L. Teo, The search for chromatically unique grahs, Grahs Combin., 6, (1990), K. M. Koh, K.L. Teo, The search for chromatically unique grahs - II, Discrete Math., 172, (1997), R. Y.Liu, A newmethodtofindthechromatic olynomial of agrahand itsalications, Kexue Tongbao, 32, (1987), R. Y. Liu, Adjoint olynomials of grahs (Chinese), J. Qinghai Normal Univ., (Natur. Sci.), 1, (1990), G. C. Lau, Y. H. Peng, Chromatic uniqueness of certain comlete t-artite grahs, Ars Comb., 92, (2009), G. C. Lau, Y. H. Peng, Chromatic uniqueness of certain comlete 4-artite grahs with certain matching or star deleted, Al. Anal. Discrete Math., 4(2), (2010), G. C. Lau, Y. H. Peng, Chromatic uniqueness of Turán grahs with certain matching or star deleted, Ars Comb., 94, (2010), R. C. Read, W. T. Tutte, Chromatic olynomials, in: L.W. Beineke, R.J. Wilson (Eds.), Selected Toics in Grah Theory, 3, Academic Press, New York, (1988), C. P. Teo, K. M. Koh, The chromaticity of comlete biartite grahs with at most one edge deleted, J. Grah Theory, 14, (1990), D. B. West, Introduction to Grah Theory, second ed., Prentice Hall, New Jersey, H. X. Zhao, Chromaticity and adjoint olynomials of grahs, Ph.D. thesis (2005) University of Twente, Netherland.