Chromatically Unique Bipartite Graphs With Certain 3-independent Partition Numbers III ABSTRACT

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1 Malaysian Chromatically Journal of Mathematical Unique Biparte Sciences Graphs with 1(1: Certain Independent (007 Partition Numbers III Chromatically Unique Bipartite Graphs With Certain 3-independent Partition Numbers III 1 Roslan Hasni & Y.H. Peng 1 Pusat Pengajian Sains Matematik Universiti Sains Malaysia, Penang, Malaysia Jabatan Matematik dan Institut Penyelidikan Matematik Universiti Putra Malaysia UPM Serdang, Malaysia hroslan@cs.usm.my, yhpeng@fsas.upm.edu.my ABSTRACT s For integers p, q, s with p q and s 0, let K ( pq denote the set of _ connected bipartite graphs which can be obtained from K(p,q by deleting a set s G K p q with p q 3, of s edges. In this paper, we prove that for any graph, and 1 s q - 1 if the number of 3-independent partitions of G is p-1 + q-1 + s + 4, then G is chromatically unique. This result extends both a theorem by Dong et al. []; and results in [4] and [5]. Keywords : Chromatic polynomial, Chromatically equivalence, Chromatically unique graphs. INTRODUCTION All graphs considered here are simple graphs. For a graph G, let V(G, E(G, δ(g, Δ(G and P(G,λ be the vertex set, edge set, minimum degree, maximum degree and the chromatic polynomial of G, respectively. Two graphs G and H are said to be chromatically equivalent (or simply χ - equivalent, symbolically G ~ H, if P(G, λ = P(H, λ. The equivalence class determined by G under ~ is denoted by [G]. A graph G is chromatically unique (or simply χ- unique if H G whenever H ~ G, i.e, [G] = {G} up to isomorphism. For a set ς of graphs, if [ G] ς for every G ς and ς of graphs, if P( G, λ P( G, λ, then ς is said to be χ closed. For two sets ς1 for every G1 ς and G 1 ς, then ς1 and 1 ς are said to be chromatically disjoint, or simply -disjoint. For integers p, q, s with p q and s 0, let K -s (p,q denote the set of connected (resp. -connected bipartite graphs which can be obtained from K p,q by deleting a set of s edges. Malaysian Journal of Mathematical Sciences 139

2 Roslan Hasni & Y.H Peng For a bipartite graph G = (A, B; E with bipartition A and B and edge set E, let be G =(A,B ;E the bipartite graph induced by the edge set where {,, } E = xy xy E x A y B A where p = A and q = B. A and B B. We write G = K( p, q G In [1], Dong et al. proved the following result. s Theorem 1.1 : For integers p, q, s with p q and 0 s q 1, ( pq closed. Throughout this paper, we fix the following conditions for p, q and s: p q 3 and 1 s q 1. κ is -, For a graph G and a positive integer k, a partition {A 1,A,...A k } of V(G is called a k-independent partition in G if each A i is a non-empty independent set of G. Let (G, k denote the number of k independent partitions in G. For any bipartite graph G = (A, B; E, define A (,3 (,3 ( 1 B G = G + 1 In [1], the authors found the following sharp bounds for (G,3. s Theorem 1.: For G κ ( p, q with p q 3 and 0 s q 1 s ( G s,3 1, where ( G,3 = s iff Δ ( G = 1 and ( G,3 = s 1 iff Δ ( G = s. s For t= 0,1,,..., let B(p, q, s, t denote the set of graphs G K ( p, q (,3 G = s+ t. Thus, K -s (p,q is partitioned into the following subsets: B(p,q,s,0, B(p,q,s,1,..,B (p,q, s, s - s - 1. Assume that B(p, q, s, t = φ for t> s s 1. with Lemma 1.1 : (Dong et al. [] For p q 3 and 0 s q 1,if 0 1 q 1 q 1, then s ( pqst,,, κ ( pq, B. 140 Malaysian Journal of Mathematical Sciences

3 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III Dong et al. [1] have shown that if G is a -connected graph in s B( p, q, s,0 B( p, q, s, s 1, then G is unique. In [], Dong et al. proved that every -connected graph in B(p, q, s, t is unique for 1 t 4. In [4] and [5], we extended this result for t = 5 and t = 6, respectively. In this paper, we prove the chromatic uniqueness of graphs in B (p, q, s, 7. PRELIMINARY RESULTS AND NOTATION For any graph G of order n, we have (see [3]: n (, λ = (, λ( λ 1...( λ + 1 P G G k k k = 1 Thus, we have Lemma.1 : If G ~ H, then (G, k = (H, k for k = 1,,... By Theorem 1.1, the following two results were obtained in []. s Theorem.1 The set ( pqst,,, κ ( pq, Corollary.1 If 0 t q 1 q 1, then ( pqst,,, B is χ closed for all t 0. B is χ closed... Let β i (G, or simply β i denote the number of vertices in G with degree i, n i (G denote the number of i-cycles in G and P n denote the path with n vertices. Then Dong et al. [] established the next two results. Lemma.: For G ( A, B; E κ s ( p, q =, i. if Δ( G, then ( G,3 s β ( ( G n4 G ii if Δ ( G = 3, then ( G,3 s β ( ( G n4 G iff N u N υ for all iii. G G Δ( G G,3 + s 1 Δ G. = + + ; + +, where equality holds u, υ B ; or u, υ B For two disjoint graphs H 1 and H, let H1 H denote the graph with vertex set V ( H and edge set E ( H ( H V H 1 1 and let kh be null if k = 0.. Let kh = 1443 H... H for k 1 k Malaysian Journal of Mathematical Sciences 141

4 Roslan Hasni & Y.H Peng s Lemma.3: Let G κ ( p, q. If (i each component of G is a path and ( (ii ( 3 G K s K 1,3 G,3 = s+ t s+ 4, then either β G = t, or Now for convenience we define the graphs Yn, Z1, Z and Z 3 as in Figure 1. Figure 1: The graphs Y n, Z 1, Z and Z 3 The following result is an extension of Lemma.3. s Lemma.4: Let G κ ( p, q. If ( G,3 = s+ 7, then either i. each component of G is a path and β (G =6, or ii. G K1,3 3P3 ( s 9 K, iii. iv. G K1,3 P5 ( s 8 K, or v. G C4 P3 ( s 8 K, vi. G C4 P4 ( s 7 K, or vii. G C6 P3 ( s 8 K, or viii. G Y3 P4 ( s 7 K, or ix. G Y3 P3 ( s 8 K, x. G Y P ( s 7 K, or or G K P P s 8 K, or 1,3 4 3 or or 4 3 G Y s 6 K, or xi. 5 G Z P s 7 K, or xii. xiii. G Z ( s 6 K xiv. G Z ( s 6 K 1 3, or 3 14 Malaysian Journal of Mathematical Sciences

5 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III Proof. Since ( G,3 = s+ 7, Δ ( G = 3 by Lemma.(iii. If.(ii, we have β ( G = 3, n4( G = 0 and β ( 3 G = 1. Δ G 3, by Lemma Thus G K 3P ( s 9 K or G K1,3 P4 P3 ( s 8 K or 1,3 3 ( 7 or G Y P ( s 7 K or ( 8 G K1,3 P5 s K 3 4 G Y P s K 3 3 or G Y P ( s 7 K or G Y5 ( s 6 Kor ( ( 6 or ( 6 G Z s K ( G n ( G 4 7 G Z P s K or If ( G G Z s K Δ =, we have β + = by Lemma.(i, and thus either G contains no cycles or only have one cycle. Hence, when Δ ( G =, either each component of G is a path, and ( β G = 7, or G K, P3 ( s 8 K, G K, P4 ( s 7 K and ( 8 G C6 P3 s K Lemma.(i. By Lemma.4, we have the following result. s Theorem. : Let G k ( p, q and ( G {,3 = s+ 7, then ( 8 ( 9 G P9 s K, P8 P3 s K, P P s 10 K, P P P s 10 K, P 3P s 10 K, P P s 9 KP P s 9 K,P P s 10 K, 7 4, P 4P s 1 K, P P P s 11 K, P P s 10 K, P P s 11 K,P 3P s 1 K, P 5P s 13 K, P s 14 K, K 3P s 9 K, K P P s 8 K, 3 1,3 3 1,3 4 3 ( 7, ( 8, ( 7 K P s K K P s K K P s K, 1,3 5, 3, 4 ( 8, ( 7, ( 8 C P s K Y P s K Y P s K, Y P s 7 K, Y s 6 K, Z P s 7 K, ( 6, ( 6 } Z s K Z s K 3 where H s i K does not exist if s < i. For a bipartite graph G = (A, B; E, let Malaysian Journal of Mathematical Sciences 143

6 Roslan Hasni & Y.H Peng Ù(G = {Q Q is an independent set in G with Q A φ, Q B φ}. For a bipartite graph G = (A, B; E, the number of 4 independent partitions {A 1,A,A 3,A 4 } in G with Ai Define A or Ai B for all i = 1,, 3,4 is ( A 1 1 ( B ( 3 A 3. A 3 ( 3 B 3. B ! 3! A = ( 1 B ( 1 A B ( ,4 =, s Observed that for G, H K ( p, q, A 1 B 1 A 1 B 1 ( G ( G 1 ( G,4 = ( H,4 iff ( G,4 = ( H,4 The following five lemmas (see [] will be used to prove our main results. s Lemma.5 : For G = ( A, B; E K ( p, q with A = p and B = q, p 1 Q A q 1Q B ( G,4 ( = + + Q Ω( G {{ Q1 Q} Q1 Q Ω G Q1 Q = /},,, 0. d u = Lemma.6 : For a bipartite graph G = ( A, B; E, if uvw is a path in G with and d ( v =, then for any k, G ( G, K = ( G + uv, k + ( G { u, v, w}, k 1 For a bipartite graph G ( A, B; E vertices in Aresp (., B with degree i. Lemma.7: For G B ( p, q, s, t, ( Q Ω ( G G 1 = let βi( G, A ( resp., β i( G, B be the number of p 1 Q A if each component of G is a path, then + q 1 Q B p q p 3 q p 3 q 3 β = s + + t + + G, A. Let p i (G denote the number of paths P i in G., if each component of G is a path, then Lemma.8 : For G B ( p, q, s, t s + t {{ Q1, Q} Q1 Q Ω( G Q1 Q = φ} = t p4 ( G p5 ( G,, Malaysian Journal of Mathematical Sciences

7 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III For G ( p q s t B,,,, define p q p q ( G,4 ( G,4 s( t( = ( s t( s t 1 3t] Observe that for GH, B ( pqst,,,, ( G,4 ( H,4 + + (1 = iff ( G,4 = ( H,4 Lemma.9: For G ( p, q, s, t, " p 3 q 3 ( G = ( β( G A p4( G p5( G B if each component of G is a path, then,4, 3. MAIN RESULT Dong et al. [1] have shown that any -connected graph G is ( pqs,,,0 ( pqs,,, s s 1 connected graph in ( pqst,,, B B is -unique. In [], Dong et al. proved that every - B is -unique for 1 t 4. In [4] and [5], we proved this result for t = 5 and t = 6 respectively. In this section, we shall prove that every - B pqst,,, is -unique for t = 7. connected graph in The following theorem is our main result: Theorem 3.1 : Let p, q and s be integers with p q 3 and 0 s q 1. For every G B ( p, q, s,7, if G is -connected, then G is -unique. s Proof : By Theorem.1, B ( pqst,,, κ ( pq, is -closed for all t 0. Hence, to show that every -connected graph in B ( pqst,,, is -unique, it suffices to show that for every two graphs G and H in B ( pqst,,,, if G H then either ( G,4 ( H,4 or ( G,5 ( H,5. Recall that ( G,4 = ( H,4 iff ( G,4 ( H,4 ( G,4 = ( H,4 iff ( G,4 ( H,4 The set ( pqs,,,7 =. = and B consists of 108 graphs by Theorem., named as G7,1, G7,, G7,3,..., G 7,108 (see Table 1 in [7]. This graphs are shown in this table with the. For each graph G7, i, if every component of values ( G " 7,1,4, ( G7,,4,... ( G7,108,4 G 7,i is a path, then ( G,4 7, i can be obtained by Lemma.9; otherwise, we must find Malaysian Journal of Mathematical Sciences 145

8 Roslan Hasni & Y.H Peng (,4 G7, i by Lemma.5, and then we find ( G,4 7, i T1 = { G7,1, G7,} T = { G7,3, G7,4,..., G7,7} T 3 = { G7,8, G7,9,..., G7,17} = T4 = { G7,18, G7,19,... G7,3} T5 = { G7,4, G7,5,..., G7,31} T6 = { G7,3, G7,33} T7 = { G7,34, G7,35,... G7,37} T8 = { G7,38, G7,39,..., G7,43} T9 = { G7,44, G7,45,... G7,53} T 10 = { G7,54, G7, 55,... G7,63} T11 = { G7,64, G7,65,... G7,67} T1 = { G7,68, G7,69,... G7,73} T13 = { G7,74, G7,75,... G7,81} T14 = { G7,8, G7,83} T15 = { G7,84, G7,85} T16 = { G7,86, G7,87,..., G7,91} T17 = { G7,9, G7,93,..., G7,97} T18 = { G7,98} T19 = { G7,99, G7,100} T 0 = { G7,101, G7,10,..., G7,104 } T 1 = { G7,105, G7,106,..., G7,108} by using Equation ( Malaysian Journal of Mathematical Sciences

9 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III For non-empty sets W 1,W,...,W k of graphs, let η ( WW,,, W = 0 if ( G1, 4 ( G, 4 every two graphs G1 Wi and G W j, where i j The values of ( G,4 7, i 1... k and let 1... for η WW,,, W = 1 otherwise. for i=5,6,7;1,13,...,17;46,47,...,53;8,83,...,108 are not given by Lemma.9, but they can be obtained by Lemma.5 and Equation (1. For example, we show a detailed computation of ( G 4 7,5, below : i. (,4 ( p q 3 3 7,5 4 ( p q ( q p G s Similarly, we obtain ( G7, i,4 = ( s p 3 q 3 s p q p 3 q s 7 s = p q for other values of i as follows (see Table 1 in [7]. p 3 q 1 ii. ( G,4 = ,6 iii. ( G7,7,4 = p 1 q 3 iv. ( G7,1 v. ( G7,13,4 = 18. p 4 q 3,4 = p 4 q vi. ( G7,14,4 = p 4 q 3 vii. ( G, ,15 = p 1 q 4 viii. ( G 7,16,4 = p 3 q 4 ix. ( G 7,17,4 = p 3 q 4 x. ( G7,46,4 = 9. p 4 xi. ( G7,47,4 = 9. p 4 q 3 xii. ( G7,48,4 = 3 9. p 4 q xiii. ( G7,49,4 = p 4 q 3 xiv. ( G7,50,4 = 9 9. p 1 q 4 xv. ( G7,51,4 = p 3 q 4 k Malaysian Journal of Mathematical Sciences 147

10 xvi. ( G7,5 p 3 q 4 xvii. ( G7,53 Roslan Hasni & Y.H Peng,4 = ,4 = p 3 q 4 xiix. ( G7,8,4 = p 4 q 3 xix. ( G7,83,4 = p 1 q 4 xx. ( G, ,84 = p 4 q xxi. ( G7,85,4 = p 3 q 4 xxii. ( G7,86,4 = 3 8. p 4 q xxiii. ( G7,87,4 = p 4 q 3 xxiv. ( G, ,88 = p 1 q 4 xxv. ( G7,89,4 = p 3 q 4 xxvi. ( G, ,90 = p 4 q xxvii. ( G7,91,4 = p 3 q 4 xxviii. ( G7,9,4 = p 3 q 3 xxix. ( G7,93,4= p 3 q 3 xxx. ( G7,94,4 = 4. p 4 q 3 xxxi. ( G7,95 xxxii. ( G7,96 xxxiii. ( G7,97 xxxiv. ( G7,98,4 = p 4 q,4 = p 3 q 4,4 = 6 4. p 3 q 4,4 = 3. p 3 q 1 xxxv. ( G, ,99 = p 4 q xxxvi. ( G7,100,4 = p 3 q 4 xxxvii. ( G, ,101 = p 4 q 3 xxxviii. ( G7,10,4 = p 4 q xxxix. ( G7,103,4 = p 3 q 4 xxxx. ( G7,104, 4 = p 3 q 4 xli. ( G7,105,4 = 1. p 4 q Malaysian Journal of Mathematical Sciences

11 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III xlii. ( G7,106 xliii. ( G7,107 xliv. ( G7,108,4 = 3 1. p 4 q,4 = p 3 q 4 p 3 q 4,4 = Claim 1. η(t 1,T,T 3,...,T 1 =0: Proof of Claim 1. Note that if k (k is an integer 1 is not a factor of x, then h is also not a factor of x for any integer h k. Similarly, if k (k is an integer 1 is a factor of x, then h is also a factor of x for any integer 1 h k. a. For s=6,t 14,T 15 and T 16 are non-empty. Observe that 3 is a factor of G, for 14 G T15 T 16. Hence, b. For s=6, 3 is a factor of ( G,4 34 G T but 3 is not a factor of η T T T =. 14, ( G, for G T 16. Hence, η ( 16 = c. For s 7, ( G,4 G, for + for G T 15 but 3 is not a factor of T T. 15, 0 is odd if G T1 T T4 T6 T7 T9 T10 T1 T14 T19 and even if G T3 T5 T8 T11 T13 T15 T16 T17 T18 T0 T 1. Hence ( T T T T T T T T T T T T T T T T T16 T T18 T T η = d. For s 7, 3 is a factor of factor of ( G,4 39 G, for G T1 T4 T10 T 14 but 3 is not + for G T T6 T7 T9 T1 T 19. Hence η ( T1 T4 T10 T14, T T6 T7 T9 T1 T 19 = 0. e. For s 7, 6 4 is a factor of ( G,4 + 3 for G T 1, is not a factor of G,4 + 3 for, G T and 4 is a factor of G T T. Hence η ( TTT 1, 4, 10 T 14 = 0. f. For s 7, 6 is a factor of ( G,4 + 7for G T 14. Hence g. For s 7, 4 is a factor of ( G,4 1 G,4 + 3 but 6 is not for G,4 + 7 for G T 10 but 6 is not a factor of Gin η T T =. 10, for G,4 + 1 for 7 9 1, but 4 is not for 6. 5 h. For 7, 3 G T19, is not a factor of T T T T and 3 is a factor of G T Hence s is a factor of G, for, for G 7 1 G 9 19, G,4 + 1 η T T T T T T =. G, for 3 G T is not a factor of, T and 3 is a factor of T T. Hence η T, T, T T = G, but 5 is not Malaysian Journal of Mathematical Sciences 149

12 Roslan Hasni & Y.H Peng i. For s 7, 4 is a factor of ( G, for G T. Hence j. For s 7, 3 is a factor of ( G,4 34 factor of ( G,4 34 G, for G T 7 but is not a factor of η T, T = 0. + for G T3 T8 T 15 but 3 is not + for G T 5 T. 11 T 13 T 16 T17 T T0 T 1 Hence ( T T T T T T T T T T T η = k. For s 7, is a factor of ( G,4 18 G, for, 8 + for (, 4 3 G T and 4 is a factor of G T. 15 Hence η ( T3, T8, T 15 = 0. l. For s 7, 3 is a factor of s 7, 3 for is not a factor of 18 G T is not a factor of G, but 5 is not for G,4 + 8 for G T16 T 1 but 3 G,4 + 8 for G T T T T T T. Hence ( m. For s 7, 4 is not factor of ( G,4 1 η T T, T T T T T T = 0. + for G T 1 but it is a factor of ( G,4 + 1 for G T. 16 Hence η ( T16, T 1 = 0. n. For s 7, 4 3 is a factor of ( G,4 + 4 for G T, 17 is not a factor of ( G,4 + 4 for G T T T, and is a factor of ( G,4 + 4 but 4 is not for G T T. Hence 13 0 η T, T T T, T T = 0. o. For s 7, 5 is a factor of G,4 + for, 5 G T. Hence 11 η T5, T11, T 18 = 0. p For s 7, 5 is a factor of ( G,4 ( G,4 for G,4 + for G T is not a factor of, 4 18 G T and 4 is a factor of G,4 + but 5 is not for for G T 13 but 5 is not a factor of G T Hence η T, T = By (a to (p, Claim 1 holds. The remaining work is to compare every two graphs in each T i except T 18 since it contains only one graph. Since this comparison process is standard, long and rather repetitive, we shall not discuss all here. In the following we only show the detail comparisons of every two graphs in T i for i = 1, and 3. The reader may refer to [6] for complete comparisons. 1. T i 1.1. When p = qg, 7,1 G7,. 150 Malaysian Journal of Mathematical Sciences

13 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III 1.. When p>q from Table 1 [7], we can easily see that ( G7,1 ( G7,. T.1. When p = q, G7,3 G7,4, G7,5 G7,7 and Thus, we have ( G7,5, 4 ( G7,6, 4 ( p 3 q ( 3 p q 19 = = + = p 3 q 3 0,,4,4 < 0. ( G7,5, 4 ( G7,3, 4 ( p 3 q ( 3 p 3 q = 19 = p 3. ( G7,5 = ( G7,6 ( G7,3,4,4 <,4. Since ( G7,5,4 = ( G7,6,4, we need to calculate ( G7,5 ( G7,6 using Lemma.6, we have ( G7,5,5 ( G7,6,5,5,5. By ( G7,5 ab 1 1,5 ( G7,5 { a1, b1},4 ( G7,5 { a1, b1, c1},4 = + + { } ( { } G 7,6 + ab 1 1,5 + G7,6 ab 1 1,4 + G7,6 a1, b1, c1,4 ( G7,5 { a1 b1 c1} G7,6 { a1 b1 c1} =,,,4,,,4 since G ab G ab + + and { } { G } 7,5 a1, b1 G7,6 ab 1 1 7, ,6 1 1, ( G7,5 { a1 b1 c1} G7,6 { a1 b1 c1} =,, 4,, 4. Since G { a, b, c } ( p, q 1, s,6 by Lemma.5, we have 7, B and { } G7,6 a1, b1, c1 B p 1, q, s, 6, ( G7,5,5 ( G7,6,5 ( G7,5 { a1 b1 c1} G7,6 { a1 b1 c1} =,,,4,,,4 Malaysian Journal of Mathematical Sciences 151

14 Roslan Hasni & Y.H Peng p 4 q 3 p 5 q 3 p 4 q 4 ( s ( 3( ( = Thus, we have p 5 q 4 s 8 ( s p 3 q 4 p 3 q 5 ( s ( ( p 4 q 5 s 8 ( s q ( s ( s ( s p 5 5 = = q-5 (since p=q. (. When p > q ( G 7,5 > ( G 7,6,5,5.. ( G7,5,4 > ( G7,6,4 ( p 3 q ( 3 p q = 19 = + < p 3 q 3 0, ( G7,6,4 ( G7,3,4 ( 3 p 3 q ( 3 p 3 q = 19 = < q 3 0, ( G7,6,4 ( G7,7,4 ( 3 p 3 q ( p 5 q = 19 = + < p 3 q 3 0, ( G7,3,4 ( G7,4,4 = + < p 3 q 3 0, ( G7,7,4 ( G7,4,4 ( p 1 5 q ( 4 p 4 q = 19 = < q 3 0, 15 Malaysian Journal of Mathematical Sciences

15 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III ( G7,3,4 ( G7,7,4 ( 3 p 3 3 q ( p 5 q = = + = 0, if < 0, if p 3 q p = q+ 1; p q+. Therefore, if p q +, we have ( G7,5 < ( G7,6 < ( G7,3 < ( G7,7 < ( G7,4,4,4,4,4,4 ; And if p=q+1, we have ( G 7,5,4 G7,6,4 G7,3,4 = G7,7,4 ( G7,4,4 Since ( G7,3,4 ( G7,7,4 0 ( G7,3 ( G7,7 < < <. = for the case p=q+1 we need to find,5,5. Since Lemma.6 cannot be used to determine ( G7,3,5 ( G7,7,5, we shall use direct counting. We first need the following definitions. For a graph G and x V ( G, N x or simply N(x denote the set of vertices y let G such that xy E( G G. Since s q 1 p 1, there exist x A and y B such that = and N x B N y = A. Thus, for any 5-independent partition {A 1,A,A 3,A 4,A 5 }, G there are at least two A i and A j with Ai A and Aj B. This means that G has only four types of 5-independent partition. We call G has partition of type k for k=0,1,,3, if there are exactly k A i s with A i G the case for k=1, and 3. Recalled that for GH, K s ( pq,, ( G,5 ( H,5 ( G = ( H,5,5. Let i ( G,5 7,3 and i ( G 7,7,5 where 1 i 3. Ω. For our purpose, we only need to consider = iff denote the number of 5-independent partition of type i, Observe that for GH, κ s ( pq,, ( G,5 ( H,5 ( G ( H = iff,5 =,5. By direct counting, we obtained the following : i. i( G7,3,5 i( G7,7,5 Malaysian Journal of Mathematical Sciences 153

16 Roslan Hasni & Y.H Peng 1 q q q q = q 1 3 ( 3 q 1 q q + ( 1( q ( 1 q q ( 3 3 q q q q q q 3 q = q 5 q 3 3. ii. ( G7,3,5 ( G7,7,5 q 4 q 3 q ( s = ( q 3 q ( q q ( q q ( s 1( q q ( s 4( q q = + q iii. 3( G7,3,5 3( G7,7,5 s 9 s ( s = 7 4( s = s 18. Thus, we have ( G7,3,5 ( G7,7,5 ( ( 1 G7,3,5 1 G7,7,5 G7,3,5 ( G7,7,5 ( 3( G7,3,5 3( G7,7,5 ( q 5 3 q 3 5 ( 34 q 6 ( s 18 = + + = q 5 q 5 q 3 = s T When p = q, G7,8 G7,9, G7,10 G7,11, G7,1 G7,17, G7,13 G7,16, G7,14 G7,15 and 154 Malaysian Journal of Mathematical Sciences

17 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III Thus we have ( G7,8 ( G7,10,4,4 = 0 ( ( G7,8,4 ( G7,1,4 ( 3 p 3 3 q ( p q 18 = p 4 = >0, ( G7,1,4 ( G7,13,4 ( p 4 q ( 3 p q 18 = p 3 q 3 = + (3 = 0, ( G7,1,4 ( G7,14,4 ( p 4 q ( 5 p 3 q 18 = = + = p q 0. ( G7,1 = ( G7,13 = ( G7,14 ( G7,8 = ( G7,10,4,4,4 <,4,4. Hence, we need to compare ( G,5 7,8 with ( G,5 7,10 ( G7, i,5, where 1 i 14 By using Lemma.6, we have ( G7,8,5 ( G7,10,5 and for each pair of ( G7,8 uv 1 1,5 ( G7,8 { u1, v1},4 ( G7,8 { u1, v1, w1},4 = ( G7,10 + uv,5 + ( G7,10 { u, v},4 + ( G7,10 { u, v, w },4 ( G7,8 { u1 v1} G7,8 { u1 v1 w1} =,,4 +,,,4 ( G7,10 { u v} + G7,10 { u v w },,4,,,4 since G7,8 + u1v1 G7,10 + ub ( G7,8 { uv 1 1} G7,10 { u v} =,4,,4 + ( G7,8 { u1 v1 w1} G7,10 { u v w},,,4,,,4 Malaysian Journal of Mathematical Sciences 155

18 Roslan Hasni & Y.H Peng ( G7,8 { u1 v1} G7,10 { u v},,4,,4 = + ( G7,8 { u1 v1 w1} G7,10 { u v w},,,4,,,4. Since {, } B ( 1, 1,,5, { } B G u v p q s 7,8 1 1 G u, v p 1, q 1, s,5, 7,10 {,, } B (, 1, 3,4, { } B G u v w p q s 7, G u, v, w p, q 1, s 3,4, 7,10 and by Lemma.9, we have ( G7,8 { u1 v1} G7,10 { u v},,4,,4 ( G7,8 { u1 v1} G7,10 { u v} =,,4,,4 ( p ( q ( p ( q = = 1, and ( G7,8 { u1 v1 w1} G7,10 { u v w},,,4,,,4 ( G7,8 { u1 v1 w1} G7,10 { u v w} =,,,4,,,4 ( p ( q ( p ( q = 6 11 = 5. Thus, we have ( G7,8 ( G7,10,5,5 = 6> 0. (4 Similarly, by Lemma.6, we have ( G7,1,5 ( G7,13,5 ( G7,1 ab, 5 ( G7,1 { a, b},4 ( G7,1 { a, b, c},4 = ( G ( { } ( { } 7,13 + ab,5 + G7,13 ab,4 + G7,13 a, b, c,4 156 Malaysian Journal of Mathematical Sciences

19 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III ( G7,1 { a b c} G7,13 { a b c} =,,,4,,,4 Since G { } { 7,1 + ab G7,13 + ab, G7,1 a, b G7,13 a, b} ( G7,1 { a b c} G7,13 { a b c} =,,,4,,,4. Since G { a, b, c } ( p, q 1, s,6 7,1 B and { } B G a, b, c p 1, q, s,6, 7,13 by Lemma.5, we have ( G7,1 { a b c} G7,13 { a b c},,,4,,,4 ( G7,1 { a b c} G7,13 { a b c} =,,,4,,,4 p 4 q 3 p 5 q 3 p 4 q 4 ( s ( 4( ( = s 8 p 6 q 3 ( s p 3 q 4 p 4 q 4 p 3 q 5 ( s ( 4( ( s p 5 q 4 ( s q ( s ( s q 6 5 = , (since p = q = > q Similarly, by Lemma.6, we have ( G7,13,5 ( G7,14,5 ( G7,13 a3b3, 5 ( G7,13 { a3, b3 },4 ( G7,13 { a3, b3, c3},4 ( G { } 7,14 + a3b3,5 + G7,14 a3, b3,4 + G7,14 a3, b3, c3,4 = ( { } ( G7,13 { a3, b3, c3} G7,14 { a3 b3 c3} =,4,,,4 Malaysian Journal of Mathematical Sciences 157

20 since Roslan Hasni & Y.H Peng G + a b G + a b, 7,13 3, 3 7, and G7,13 { a3, b3} G7,14 { a3, b3} ( G7,13 { a3, b3, c3},4 ( G7,14 { a3, b3, c3},4. = Since G { a b c } ( p q s 7,13 3, 3, 3 B, 1,, 6 and { } B G a, b, c p 1, q, s,6, 7, by Lemma.5 we have,4 (,,,4 ( G7,13 { a3, b3, c3},4 ( G7,14 { a3, b3, c3},4 ( G7,13 { a3, b3, c3} G7,14 { a3 b3 c3} = p 4 q 3 p 5 q p 4 q 4 ( s ( 3( 3 ( = s 8 p 6 q ( { 1( s 8 30 } p 3 q 4 p 4 q 4 p 3 q 5 ( s ( 3( ( s 8 p 5 q 4 ( s q ( s ( s q 6 5 = (since p = q = > q When p>q from Equation (3, we have ( G7,1, ( G7,13, ( G7,13, 4 ( G7,14, = + < 0, p 3 q 3 p ( 4 q p 3 18 ( 5 4 q = = + < p 3 q 3 0, 158 Malaysian Journal of Mathematical Sciences

21 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III ( G7,14, 4 ( G7,8, 4 p ( 4 q 3 p ( 3 3 q = = < p 4 0, From Equation (, we have ( G7,8, ( G7,10, ( G7,8, 4 ( G7,15, = 0, p ( 3 q 3 p ( 1 q = = + < p 3 q 4 3 0, ( G7,15, 4 ( G7,9, 4 ( p 1 9 q ( 4 p 4 q = 18 = < p q 4 0, ( G7,9, ( G7,11, ( G7,9, 4 ( G7,16, 4 Therefore, 4 4 = 0, p ( 3 q 3 p ( 5 3 q = = + < p 3 q 4 3 0, ( G7,16, 4 ( G7,17, 4 p ( 3 q 4 p ( 6 3 q = = + < p 3 q 3 0. ( G7,1,4 ( G7,13,4 ( G7,14,4 ( G7,8,4 = ( G7,10,4 < < < < ( G7,15 < ( G7,9 = ( G7,11 < ( G7,16 < ( G7,17,4,4,4,4,4. Malaysian Journal of Mathematical Sciences 159

22 Roslan Hasni & Y.H Peng Since ( G7,8,4 = ( G7,10,4 and ( G7,9 ( G7,11 ( G7,8,5 ( G7,10,5 and ( G7,9 ( G7,11 have ( G7,8,5 ( G7,10,5 6 0.,4 =,4, we need to calculate,5,5. Hence,by Equation (4, we = > By using Lemma.6, we have ( G7,9,5 ( G7,11,5 ( G7,9 u3v3,5 ( G7,9 { u3v3},4 ( G7,9 { u3, v3, w3},4 = ( G7,11 + u4v4,5 + ( G7,11 { u4, v4},4 + ( G7,11 { u4, v4, w4},4 ( G7,9 { u3, v3},4 ( G7,9 { u3, v3, w3},4 = + ( G7,11 { u4, v4},4 + ( G7,11 { u4, v4, w4},4 since G7,9 + u3v3 G7,11 + u4, v4 ( G7,9 { u3, v3},4 ( G7,11 { u4, v4},4 = + ( G7,9 { u3 v3 w3} G7,11 { u4 v4 w4},,,4,,,4 + ( G7,9 { u3 v3} G7,11 { u4 v4} =,,4,,4 + ( G7,9 { u3 v3 w3} G7,11 { u4 v4 w4},,,4,,,4. Since G { u, v } ( p 1, q 1, s,5 7,9 3 3 B and { } B G u, v p 1, q, s,5, 7, by Lemmas.5,.7 and.8, we have ( G7,9 { u3 v3} G7,11 { u4 v4},,4,,4 ( G7,9 { u3, v3},4 ( G7,11 { u4, v4},4 p 3 q 3 p 4 q 3 p 4 q 4 ( s ( 5( 3( = = Malaysian Journal of Mathematical Sciences

23 Chromatically Unique Biparte Graphs with Certain 3-Independent Partition Numbers III s 3 p 3 q 3 p 4 q 3 5 ( s ( + + 5( + + p 4 q 4 s = 1. Similarly, since {,, } B ( 1,, 3,4 { } B G u v w p q s and 7, G u, v, w p 1, q, s 3,4, 7, by Lemma.5,.7 and.8, we have ( G7,9 { u3, v3, w3},4 ( G7,11 { u4, v4, w4},4 ( G7,9 { u3, v3, w3},4 ( G7,11 { u4, v4, w4},4 p 3 q 4 p 4 q 4 p 4 q 5 ( s 3( 4( ( = = s p 4 q 5 s + 1 ( + 3 = 5. Thus we have ( G7,9 ( G7,11 p 3 q 4 p 4 q 4 ( s,5,5 = 6> 0. For the remaining working every two graphs in T 4 to T 1, the reader may refer to [6]. This completes the proof of Theorem 3.1. Malaysian Journal of Mathematical Sciences 161

24 Roslan Hasni & Y.H Peng REFERENCES DONG, F.M., K.M. KOH, K.L. TEO, C.H.C. LITTLE and M.D. HENDY Sharp bounds for the numbers of 3-partitions and the chromaticity of bipartite graphs. J. Graph Theory 37: DONG, F.M., K.M. KOH, K.L. TEO, C.H.C. LITTLE and M.D. HENDY Chromatically unique graphs with low 3-independent partition numbers. Discrete Math. 4: READ, R.C. and W.T. TUTTE Chromatic polynomials. In Selected Topics in Graph Theory III, ed. L.W. Beineke and R.J. Wilson, p New York: Academic Press. ROSLAN, H. and Y.H. PENG. Chromatically unique bipartite graphs with certain 3- independent partition numbers. Matematika (accepted for publication. ROSLAN, H. and Y.H. PENG. Chromatically unique bipartite graphs with certain 3- independent partition numbers II. Bull. Malaysian Maths Soc. (accepted for publication. sas.upm.edu.my/~yhpeng/publish/proof t7. pdf sas.upm.edu.my/~yhpeng/publish/tablet7.pdf 16 Malaysian Journal of Mathematical Sciences