Sierpi nski graphs as spanning subgraphs of Hanoi graphs Sara - PowerPoint PPT Presentation

Definitions Main result Sierpi´ nski graphs as spanning subgraphs of Hanoi graphs Sara Sabrina Zemljiˇ c Andreas M. Hinz Sandi Klavˇ zar Institute of Mathematics, Physics and Mechanics September 2012

Definitions Main result Hanoi graphs Tower of Hanoi puzzle 1 2 3 3 4 4 1 5 5 6 2 6 0 1 2 3 0 1 2 3 • divine rule: no larger disc is (put) on a smaller disc • regular state: distribution of disc obeying the divine rule • perfect state: regular state with all discs on one peg Hanoi graph H n p ... ... represents the Tower of Hanoi puzzle with p pegs and n discs. • vertices = regular states ⇒ V ( H n p ) = [ p ] n 0 • two vertices are adjacent if one can be obtained form the other by a legal move

Definitions Main result Hanoi graphs 03 000 001 002 00 02 01 021 012 022 011 32 31 020 010 33 122 211 12 21 120 121 212 210 30 110 101 202 220 11 22 13 23 10 20 111 112 102 100 200 201 221 222 Figure: The Hanoi graphs H 3 3 and H 2 4

Definitions Main result Sierpi´ nski graphs nski graph S n Sierpi´ p • V ( S n p ) = [ p ] n 0 • two vertices s n . . . s 1 and r n . . . r 1 are adjacent if there exists an index δ , such that (i) s ℓ = r ℓ , for ℓ = n , . . . , δ + 1; (ii) s δ � = r δ ; and (iii) s ℓ = r δ and r ℓ = s δ for ℓ = 1, . . . , δ − 1. Equivalently: E ( S 1 � { i , j } | i � = j ∈ { 0, 1 . . . , p − 1 } � p ) = � { is , ir } | i = 0, 1, . . . , p − 1 , { s , r } ∈ E ( S n − 1 E ( S n � ∪ p ) = ) p � { ij n − 1 , ji n − 1 } | i � = j ∈ { 0, 1 . . . , p − 1 } � ( n ≥ 2 )

Definitions Main result Sierpi´ nski graphs 000 01 02 00 001 002 010 020 03 011 022 30 012 021 100 200 31 33 32 101 102 201 202 10 20 11 22 13 23 110 120 210 220 111 112 121 122 211 212 221 222 12 21 nski graphs S 3 3 and S 2 Figure: The Sierpi´ 4

Definitions Main result Notations ii . . . i = i n – extreme vertex in S n p or perfect vertex in H n p – p extreme vertices in S n p – degree of any extreme vertex is p − 1, all other vertices have degree p – p perfect vertices in H n p – degree of any perfect vertex is p − 1, all other vertices have degree ≥ 2 p − 3 (for n ≥ 2) s n . . . s r + 1 S r p = { s n . . . s r + 1 s | s ∈ S r p } ≃ S r p s n . . . s r + 1 H r p = { s n . . . s r + 1 s | s ∈ H r p } ≃ H r p

Definitions Main result Theorem Theorem Let p , n ∈ N . Then S n p can be embedded isomorphically into H n p if and only if p is odd or n = 1 . • n = 1: H 1 p = S 1 p = K p • p odd: – H n 1 = S n 1 = K 1 – For p ≥ 3 define for each k ∈ [ p ] 0 π k ( i ) = 1 � � k ( p + 1 ) − i ( p − 1 ) mod p ... permutation of [ p ] 0 , 2 π n k ( s n . . . s 1 ) = π k ( s n ) . . . π k ( s 1 ) ... permutation of [ p ] n 0 . The embedding ι n + 1 : V ( S n + 1 ) → V ( H n + 1 ) is defined by p p ι n + 1 ( ks ) = k π n k ( ι n ( s )) .

Definitions Main result Proof 00 00 01 04 03 02 10 40 13 42 02 03 01 04 14 41 10 40 11 44 11 44 12 43 14 41 13 42 12 43 21 34 20 30 21 34 24 31 24 31 23 32 22 33 22 33 23 32 20 30 Figure: Isomorphic embedding ι 2 from S 2 5 into H 2 5

Definitions Main result Proof - p even Theorem Let p , n ∈ N . Then S n p can be embedded isomorphically into H n p if and only if p is odd or n = 1 . • p even: – H n 2 ... n − 1 disjoint K 2 2 = P 2 n ... path on 2 n vertices S n – p ≥ 4 Lemma Every complete subgraph of H n p , p , n ∈ N , is induced by edges corresponding to moves of one and the same disc. In particular, ω ( H n p ) = p and the only p-cliques of H n p are of the form s n . . . s 2 H 1 p .

Definitions Main result Proof - p even Theorem Let p , n ∈ N . Then S n p can be embedded isomorphically into H n p if and only if p is odd or n = 1 . • p ≥ 4 even: – n = 2: extreme to perfect vertices and p -cliques to p -cliques remaining: ( p 2 ) non-incident edges in S 2 p , but there are only p ⌊ p − 1 2 ⌋ non-incident edges in H 2 p – n ≥ 3: subgraph i n − 2 S 2 p has to be mapped to some j n − 2 H 2 p , a contradition

Definitions Main result Proof - p even 03 01 02 00 00 02 01 03 30 32 31 33 31 33 32 12 21 10 20 30 11 13 23 22 11 22 13 23 10 20 12 21 Figure: Graphs S 2 4 and H 2 4

Definitions Main result Corollary Hamming graph ... ... is the graph with vertex set [ r 1 ] × [ r 2 ] × . . . × [ r n ] , where two vertices are adjacent if they differ in precisely one coordinate. ... is the Cartesian product of complete graphs K r 1 � K r 2 � . . . � K r n . K n p = K p � K p � . . . � K p Corollary Let p be odd. Then for any n, S n p is a spanning subgraph of the Hamming graph K n p .