Complete Tripartite Graphs and their Competition Numbers
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1 Complete Tripartite Graphs and their Competition Numbers Jaromy Kuhl Department of Mathematics and Statistics University of West Florida Jaromy Kuhl (UWF) Competition Numbers 1 / 16
2 Competition Graphs Definition 1 Let D = (V, A) be a digraph. The competition graph of D is the simple graph G = (V, E) where {u, v} E if and only if N + (u) N + (v). Jaromy Kuhl (UWF) Competition Numbers 2 / 16
3 Competition Graphs Definition 1 Let D = (V, A) be a digraph. The competition graph of D is the simple graph G = (V, E) where {u, v} E if and only if N + (u) N + (v). The competition graph of D is denoted C(D). Jaromy Kuhl (UWF) Competition Numbers 2 / 16
4 Competition Graphs Definition 1 Let D = (V, A) be a digraph. The competition graph of D is the simple graph G = (V, E) where {u, v} E if and only if N + (u) N + (v). The competition graph of D is denoted C(D). Which graphs are competition graphs? Jaromy Kuhl (UWF) Competition Numbers 2 / 16
5 Let S = {S 1,..., S m } be a family of cliques in a graph G. Jaromy Kuhl (UWF) Competition Numbers 3 / 16
6 Let S = {S 1,..., S m } be a family of cliques in a graph G. S is an edge clique cover of G provided {u, v} E(G) if and only if {u, v} S i. Jaromy Kuhl (UWF) Competition Numbers 3 / 16
7 Let S = {S 1,..., S m } be a family of cliques in a graph G. S is an edge clique cover of G provided {u, v} E(G) if and only if {u, v} S i. θ e (G) = min{ S : S is an edge clique cover of G}. Jaromy Kuhl (UWF) Competition Numbers 3 / 16
8 Which graphs are competition graphs of acyclic digraphs? Jaromy Kuhl (UWF) Competition Numbers 4 / 16
9 Which graphs are competition graphs of acyclic digraphs? For sufficiently large k, G I k is the competition graph of an acyclic digraph. Jaromy Kuhl (UWF) Competition Numbers 4 / 16
10 Which graphs are competition graphs of acyclic digraphs? For sufficiently large k, G I k is the competition graph of an acyclic digraph. Definition 2 The competition number of G is k(g) = min{k : G I k is the competition graph of an acyclic digraph} Jaromy Kuhl (UWF) Competition Numbers 4 / 16
11 A digraph D = (V, A) is acyclic if and only if there is an ordering v 1, v 2,..., v n of the vertices in V such that if (v i, v j ) A, then i < j. Jaromy Kuhl (UWF) Competition Numbers 5 / 16
12 A digraph D = (V, A) is acyclic if and only if there is an ordering v 1, v 2,..., v n of the vertices in V such that if (v i, v j ) A, then i < j. G is the competition graph of an acyclic digraph if and only if there is an ordering v 1,..., v n and there is an edge clique cover {S 1,..., S n } such that S i {v 1,..., v i 1 } for each i. Jaromy Kuhl (UWF) Competition Numbers 5 / 16
13 Theorem 1 k(k n,n ) = n 2 2n + 2 Jaromy Kuhl (UWF) Competition Numbers 6 / 16
14 Theorem 1 k(k n,n ) = n 2 2n + 2 Theorem 2 k(k n,n,n ) = n 2 3n + 4 Jaromy Kuhl (UWF) Competition Numbers 6 / 16
15 Theorem 1 k(k n,n ) = n 2 2n + 2 Theorem 2 k(k n,n,n ) = n 2 3n + 4 Theorem 3 For positive integers x, y and z where 2 x y z, { yz 2y z + 4, if x = y k(k x,y,z ) = yz z y x + 3, if x y Jaromy Kuhl (UWF) Competition Numbers 6 / 16
16 Theorem 4 If n 5 is odd, then n 2 4n + 7 k(k 4 n ) n 2 4n + 8. Theorem 5 If n is prime and m n, then k(k m n ) n 2 2n + 3. Jaromy Kuhl (UWF) Competition Numbers 7 / 16
17 Theorem 6 k(k n,n,n ) = n 2 3n + 4 Proof. Let L be the latin square of order n such that (a, b, c) L if and only if c a + b 1 mod n. Consider the cliques (1, 1, 1), (2, n, 1), (1, n, n), (n, 1, n), (n, 2, 1), (1, 2, 2), and (n 1, 2, n), (2, n 1, n), (1, n 1, n 1), (n 2, 2, n 1), (2, n 2, n 1), (1, n 2, n 2),... Jaromy Kuhl (UWF) Competition Numbers 8 / 16
18 Consider K x,y,z (x y z). Jaromy Kuhl (UWF) Competition Numbers 9 / 16
19 Consider K x,y,z (x y z). Definition 3 An r-multi latin square of order n is an n n array of nr symbols such that each symbol appears once in each row and column and each cell contains r symbols. Jaromy Kuhl (UWF) Competition Numbers 9 / 16
20 K 2,4,6 Jaromy Kuhl (UWF) Competition Numbers 10 / 16
21 K 2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7 Jaromy Kuhl (UWF) Competition Numbers 10 / 16
22 K 2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7 1,2 4, ,4 1,6 5 2 Jaromy Kuhl (UWF) Competition Numbers 10 / 16
23 K 2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7 1,2 4, ,4 1,6 5 2 F = { (1, 1, 1), (1, 1, 2), (1, 2, 4), (1, 2, 5), (1, 3, 3), (1, 4, 6), (2, 1, 3), (2, 1, 4), (2, 2, 1), (2, 2, 6), (2, 3, 5), (2, 4, 2), Jaromy Kuhl (UWF) Competition Numbers 10 / 16
24 K 2,4,6 1,2 4,5 3,7 6,8 5,6 7,8 1,2 3,4 7,8 2,3 4,6 1,5 3,4 1,6 5,8 2,7 1,2 4, ,4 1,6 5 2 F = { (1, 1, 1), (1, 1, 2), (1, 2, 4), (1, 2, 5), (1, 3, 3), (1, 4, 6), (2, 1, 3), (2, 1, 4), (2, 2, 1), (2, 2, 6), (2, 3, 5), (2, 4, 2), (1, 5), (1, 6), (3, 1), (3, 2), (4, 3), (4, 4), (2, 2), (2, 3), (3, 4), (3, 6), (4, 1), (4, 5)} Jaromy Kuhl (UWF) Competition Numbers 10 / 16
25 Let q and r be positive integers such that z = qy + r. Jaromy Kuhl (UWF) Competition Numbers 11 / 16
26 Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y. Jaromy Kuhl (UWF) Competition Numbers 11 / 16
27 Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y. Let R = {r i : 1 i x} be a set of rows and let S = {s i : 1 i z} be a set of z symbols. Jaromy Kuhl (UWF) Competition Numbers 11 / 16
28 Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y. Let R = {r i : 1 i x} be a set of rows and let S = {s i : 1 i z} be a set of z symbols. Set L(R, C, S ) = {(r i, c j, s k ) : (r i, c j, s k ) L, r i R, s k S }. Jaromy Kuhl (UWF) Competition Numbers 11 / 16
29 Let q and r be positive integers such that z = qy + r. Let L be a (q + 1)-multi latin square of order y. Let R = {r i : 1 i x} be a set of rows and let S = {s i : 1 i z} be a set of z symbols. Set L(R, C, S ) = {(r i, c j, s k ) : (r i, c j, s k ) L, r i R, s k S }. Lemma 1 The family F = { (i, j, k) : (r i, c j, s k ) L(R, C, S )} { (j, k) : (r i, c j, s k ) L(R \ R, C, S )} is an edge clique cover of K x,y,z. Moreover, θ(k x,y,z ) = yz. Jaromy Kuhl (UWF) Competition Numbers 11 / 16
30 Theorem 7 For positive integers x, y and z where 2 x y z, { yz 2y z + 4, if x = y k(k x,y,z ) = yz z y x + 3, if x y Jaromy Kuhl (UWF) Competition Numbers 12 / 16
31 Theorem 7 For positive integers x, y and z where 2 x y z, { yz 2y z + 4, if x = y k(k x,y,z ) = yz z y x + 3, if x y Let L be a (q + 1)-multi latin square of order y such that (i, j, k) L if and only if i + j 1 k mod y. Jaromy Kuhl (UWF) Competition Numbers 12 / 16
32 Theorem 7 For positive integers x, y and z where 2 x y z, { yz 2y z + 4, if x = y k(k x,y,z ) = yz z y x + 3, if x y Let L be a (q + 1)-multi latin square of order y such that (i, j, k) L if and only if i + j 1 k mod y. Let R = {r 1,..., r x 1, r y } and let S = {s 1,..., s z }. Jaromy Kuhl (UWF) Competition Numbers 12 / 16
33 Example: x = 3, y = 5, z = 13 1,6,11 2,7,12 3,8,13 4,9,14 5,10,15 2,7,12 3,8,13 4,9,14 5,10,15 1,6,11 3,8,13 4,9,14 5,10,15 1,6,11 2,7,12 4,9,14 5,10,15 1,6,11 2,7,12 3,8,13 5,10,15 1,6,11 2,7,12 3,8,13 4,9,14 Jaromy Kuhl (UWF) Competition Numbers 13 / 16
34 Example: x = 3, y = 5, z = 13 1,6,11 2,7,12 3,8,13 4,9,14 5,10,15 2,7,12 3,8,13 4,9,14 5,10,15 1,6,11 3,8,13 4,9,14 5,10,15 1,6,11 2,7,12 4,9,14 5,10,15 1,6,11 2,7,12 3,8,13 5,10,15 1,6,11 2,7,12 3,8,13 4,9,14 1,6,11 2,7,12 3,8,13 4,9 5,10 2,7,12 3,8,13 4,9 5,10 1,6,11 5,10 1,6,11 2,7,12 3,8,13 4,9 Jaromy Kuhl (UWF) Competition Numbers 13 / 16
35 Case 1: x = y 1 = {u 1, v 1, w 1 }, 2 = {u 2, v y, w 1 }, 3 = {u 1, v y, w y }, 4 = {u y, v 1, w y }, 5 = {u y, v 2, w 1 }, 6 = {u 1, v 2, w 2 } Jaromy Kuhl (UWF) Competition Numbers 14 / 16
36 Case 1: x = y 1 = {u 1, v 1, w 1 }, 2 = {u 2, v y, w 1 }, 3 = {u 1, v y, w y }, 4 = {u y, v 1, w y }, 5 = {u y, v 2, w 1 }, 6 = {u 1, v 2, w 2 } 0 s y 4: 3s+7 = {u y s 1, v 2, w y s }, 3s+8 = {u 2, v y s 1, w y s }, and 3s+9 = {u 1, v y s 1, w y s 1 } Jaromy Kuhl (UWF) Competition Numbers 14 / 16
37 Case 1: x = y 1 = {u 1, v 1, w 1 }, 2 = {u 2, v y, w 1 }, 3 = {u 1, v y, w y }, 4 = {u y, v 1, w y }, 5 = {u y, v 2, w 1 }, 6 = {u 1, v 2, w 2 } 0 s y 4: 3s+7 = {u y s 1, v 2, w y s }, 3s+8 = {u 2, v y s 1, w y s }, and 3s+9 = {u 1, v y s 1, w y s 1 } 0 s z y 1: 3y 2+s = {u, v, w y+s+1 } Jaromy Kuhl (UWF) Competition Numbers 14 / 16
38 Case 2: x < y 1 = {u x, v 1, w y }, 2 = {v 2, w y }, 3 = {u x, v 2, w 1 }, 4 = {u 1, v 1, w 1 }, 5 = {u 1, v 2, w 2 }, 6 = {u 2, v 1, w 2 }, 7 = {u 2, v y, w 1 } Jaromy Kuhl (UWF) Competition Numbers 15 / 16
39 Case 2: x < y 1 = {u x, v 1, w y }, 2 = {v 2, w y }, 3 = {u x, v 2, w 1 }, 4 = {u 1, v 1, w 1 }, 5 = {u 1, v 2, w 2 }, 6 = {u 2, v 1, w 2 }, 7 = {u 2, v y, w 1 } 0 s y x 1: 2s+8 = {u x, v y s, w y s 1 }, and 2s+9 = {u 1, v y s 1, w y s 1 } Jaromy Kuhl (UWF) Competition Numbers 15 / 16
40 0 s x 4: 3s+2(y x)+8 = {u x s 1, v 2, w x s }, 3s+2(y x)+9 = {u 2, v x s 1, w x s }, and 3s+2(y x)+10 = {u 1, v x s 1, w x s 1 } 0 s z y 1: 2y+x 1+s = {u, v, w y+s+1 } Jaromy Kuhl (UWF) Competition Numbers 16 / 16
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