Designs and i -Block-Intersection Graphs David Pike Memorial University of Newfoundland
Definition: A combinatorial design D consists of a set V of elements (called points), together with a set B of subsets (called blocks) of V . A balanced incomplete balanced design, BIBD ( v, k, λ ) , is a design in which: • | V | = v , • for each block B ∈ B , | B | = k , and • each 2-subset of V occurs in precisely λ blocks of B . A BIBD ( v, 3 , 1) is a Steiner triple system, STS ( v ) . A BIBD ( v, 3 , 2) is a twofold triple system, TTS ( v ) . Slide 2 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) v = 13 k = 3 λ = 1 ( k 2 ) Slide 3 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) r = λ ( v − 1) v = 13 k = 3 λ = 1 = 6 ( k 2 ) k − 1 Slide 3 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) λ ( v 2 ) r = λ ( v − 1) v = 13 k = 3 λ = 1 = 6 b = 2 ) = 26 ( k 2 ) ( k k − 1 Slide 3 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) λ ( v 2 ) r = λ ( v − 1) v = 13 k = 3 λ = 1 = 6 b = 2 ) = 26 ( k 2 ) ( k k − 1 1 0 2 C 3 B 4 A 5 9 6 8 7 Slide 3 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) λ ( v 2 ) r = λ ( v − 1) v = 13 k = 3 λ = 1 = 6 b = 2 ) = 26 ( k 2 ) ( k k − 1 { 0,2,7 } { 8,9,C } 1 0 2 C 3 B 4 A 5 9 6 8 7 Slide 3 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) λ ( v 2 ) r = λ ( v − 1) v = 13 k = 3 λ = 1 = 6 b = 2 ) = 26 ( k 2 ) ( k k − 1 { 0,2,7 } { 8,9,C } 1 0 2 { 1,3,8 } { 9,A,0 } C 3 B 4 A 5 9 6 8 7 Slide 3 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) λ ( v 2 ) r = λ ( v − 1) v = 13 k = 3 λ = 1 = 6 b = 2 ) = 26 ( k 2 ) ( k k − 1 { 0,2,7 } { 8,9,C } 1 0 2 { 1,3,8 } { 9,A,0 } { 2,4,9 } { A,B,1 } C 3 B 4 A 5 9 6 8 7 Slide 3 of 32
Example: a BIBD(13,3,1) ... i.e. , a STS(13): λ ( v 2 ) λ ( v 2 ) r = λ ( v − 1) v = 13 k = 3 λ = 1 = 6 b = 2 ) = 26 ( k 2 ) ( k k − 1 { 0,2,7 } { 8,9,C } 1 0 2 { 1,3,8 } { 9,A,0 } { 2,4,9 } { A,B,1 } C 3 { 3,5,A } { B,C,2 } { 4,6,B } { C,0,3 } { 5,7,C } { 0,1,4 } B 4 { 6,8,0 } { 1,2,5 } { 7,9,1 } { 2,3,6 } A 5 { 8,A,2 } { 3,4,7 } { 9,B,3 } { 4,5,8 } { A,C,4 } { 5,6,9 } 9 6 { B,0,5 } { 6,7,A } 8 7 { C,1,6 } { 7,8,B } This is equivalent to a K 3 -decomposition of K 13 Slide 3 of 32
Example: a TTS(6) ... i.e. , a BIBD(6,3,2): V = { 1 , 2 , 3 , 4 , 5 , 6 } Blocks: {1,2,3} {2,3,5} {1,2,4} {2,4,6} {1,3,6} {2,5,6} {1,4,5} {3,4,5} {1,5,6} {3,4,6} This is equivalent to a K 3 -decomposition of 2 K 6 Slide 4 of 32
Example: a TTS(6) ... i.e. , a BIBD(6,3,2): V = { 1 , 2 , 3 , 4 , 5 , 6 } Blocks: {1,2,3} {2,3,5} {1,2,4} {2,4,6} {1,3,6} {2,5,6} {1,4,5} {3,4,5} {1,5,6} {3,4,6} This is equivalent to a K 3 -decomposition of 2 K 6 Theorem A STS ( v ) exists if and only if v ≡ 1 or 3 (mod 6). A TTS ( v ) exists if and only if v ≡ 0 or 1 (mod 3). Slide 4 of 32
Applications of Designs: Design of experiments: Suppose we need to test how v varieties of something interact in pairs ( e.g. , crop varieties, strains of bacteria, electronic components, etc. ). Slide 5 of 32
Applications of Designs: Design of experiments: Suppose we need to test how v varieties of something interact in pairs ( e.g. , crop varieties, strains of bacteria, electronic components, etc. ). � k � Suppose also that we are able to test all pairs of any 2 k -subset of varieties in parallel. Slide 5 of 32
Applications of Designs: Design of experiments: Suppose we need to test how v varieties of something interact in pairs ( e.g. , crop varieties, strains of bacteria, electronic components, etc. ). � k � Suppose also that we are able to test all pairs of any 2 k -subset of varieties in parallel. Testing a k -subset is costly ( e.g. , each test corresponds to a crop to plant and grow, a petri dish to prepare and incubate, or a prototypic circuit board to be built). So we wish to minimise the number of tests that are necessary. Slide 5 of 32
Applications of Designs: Design of experiments: Suppose we need to test how v varieties of something interact in pairs ( e.g. , crop varieties, strains of bacteria, electronic components, etc. ). � k � Suppose also that we are able to test all pairs of any 2 k -subset of varieties in parallel. Testing a k -subset is costly ( e.g. , each test corresponds to a crop to plant and grow, a petri dish to prepare and incubate, or a prototypic circuit board to be built). So we wish to minimise the number of tests that are necessary. Solution: use a BIBD ( v, k, 1) to determine which k elements will comprise each test. Slide 5 of 32
Applications of Designs: Team Building: Suppose a professor has v graduate students who are to be sent to work at a field station, k at a time, and so that each pair of students works together λ times. Slide 6 of 32
Applications of Designs: Team Building: Suppose a professor has v graduate students who are to be sent to work at a field station, k at a time, and so that each pair of students works together λ times. How can the groups of k students to be formed? Slide 6 of 32
Applications of Designs: Team Building: Suppose a professor has v graduate students who are to be sent to work at a field station, k at a time, and so that each pair of students works together λ times. How can the groups of k students to be formed? Solution: use a BIBD ( v, k, λ ) . Slide 6 of 32
Applications of Designs: Team Building: Suppose a professor has v graduate students who are to be sent to work at a field station, k at a time, and so that each pair of students works together λ times. How can the groups of k students to be formed? Solution: use a BIBD ( v, k, λ ) . What about Scheduling? How can travel costs to/from the field station be minimised? Are some orderings of the crops/bacteria/electronics to be tested better than other orderings? How can we nicely order the blocks of a design? Slide 6 of 32
Definition: Given a combinatorial design D with block set B , the block-intersection graph of D is the graph having B as its vertex set, and in which two vertices B 1 and B 2 are adjacent if and only if B 1 ∩ B 2 � = ∅ . Slide 7 of 32
Definition: Given a combinatorial design D with block set B , the block-intersection graph of D is the graph having B as its vertex set, and in which two vertices B 1 and B 2 are adjacent if and only if B 1 ∩ B 2 � = ∅ . Example: A TTS(4): V = { 1 , 2 , 3 , 4 } � � B = { 1 , 2 , 3 } , { 1 , 2 , 4 } , { 1 , 3 , 4 } , { 2 , 3 , 4 } 123 124 234 134 Slide 7 of 32
Question (Graham, 1987) Is the block-intersection graph of a STS ( v ) Hamiltonian? Slide 8 of 32
Some Subsequent Discoveries: • BIBD ( v, k, λ ) ⇒ Hamiltonian (Horák and Rosa, 1988) • PBD ( v, K, 1) with max K � 2 min K ⇒ Hamiltonian (Alspach, Heinrich and Mohar, 1990) Slide 9 of 32
Some Subsequent Discoveries: • BIBD ( v, k, λ ) ⇒ Hamiltonian (Horák and Rosa, 1988) • PBD ( v, K, 1) with max K � 2 min K ⇒ Hamiltonian (Alspach, Heinrich and Mohar, 1990) • BIBD ( v, k, 1) with k � 3 ⇒ edge-pancyclic (Alspach and Hare, 1991) • PBD ( v, K, 1) with min K � 3 ⇒ edge-pancyclic (Hare, 1995) Slide 9 of 32
Some Subsequent Discoveries: • BIBD ( v, k, λ ) ⇒ Hamiltonian (Horák and Rosa, 1988) • PBD ( v, K, 1) with max K � 2 min K ⇒ Hamiltonian (Alspach, Heinrich and Mohar, 1990) • BIBD ( v, k, 1) with k � 3 ⇒ edge-pancyclic (Alspach and Hare, 1991) • PBD ( v, K, 1) with min K � 3 ⇒ edge-pancyclic (Hare, 1995) • BIBD ( v, k, λ ) ⇒ pancyclic (Mamut, Pike and Raines, 2004) • PBD ( v, K, λ ) with max K � λ min K ⇒ pancyclic (Case and Pike, 2008) Slide 9 of 32
Some Subsequent Discoveries: • BIBD ( v, k, λ ) ⇒ Hamiltonian (Horák and Rosa, 1988) • PBD ( v, K, 1) with max K � 2 min K ⇒ Hamiltonian (Alspach, Heinrich and Mohar, 1990) • BIBD ( v, k, 1) with k � 3 ⇒ edge-pancyclic (Alspach and Hare, 1991) • PBD ( v, K, 1) with min K � 3 ⇒ edge-pancyclic (Hare, 1995) • BIBD ( v, k, λ ) ⇒ pancyclic (Mamut, Pike and Raines, 2004) • PBD ( v, K, λ ) with max K � λ min K ⇒ pancyclic (Case and Pike, 2008) • BIBD ( v, k, λ ) with k � 3 ⇒ cycle extendible (Abueida and Pike, 2013) • PBD ( v, K, λ ) with λ � 2 and max K � λ (min K − 1) ⇒ cycle extendible (Luther and Pike, 201x) Slide 9 of 32
Definition: Given a combinatorial design D with block set B , the i -block-intersection graph of D is the graph having B as its vertex set, and in which two vertices B 1 and B 2 are adjacent if and only if | B 1 ∩ B 2 | = i . Slide 10 of 32
Definition: Given a combinatorial design D with block set B , the i -block-intersection graph of D is the graph having B as its vertex set, and in which two vertices B 1 and B 2 are adjacent if and only if | B 1 ∩ B 2 | = i . Observations: When i � 1 , the i -block-intersection graph of D is a subgraph of the traditional block-intersection graph. However, when i = 0 , the 0-block-intersection graph of D is the graph complement of the traditional block-intersection graph. Slide 10 of 32
Example: Our previous TTS(6) The graph 123 235 136 124 346 246 256 345 156 145 Slide 11 of 32
Example: Our previous TTS(6) The traditional block-intersection graph: 123 235 136 124 346 246 256 345 156 145 Slide 11 of 32
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