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 1 2 3 4 5 6 1 2 3 1 2 3 4 5 6

• 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 Hn

p ...

... represents the Tower of Hanoi puzzle with p pegs and n discs.

• vertices = regular states ⇒ V (Hn

p) = [p]n

• two vertices are adjacent if one can be obtained form the
• ther by a legal move

Definitions Main result

Hanoi graphs

000 001 002 021 022 020 012 010 011 122 120 121 110 111 112 101 102 100 211 212 210 202 200 201 220 221 222 03 02 01 00 12 13 10 11 21 20 23 22 31 32 30 33

Figure: The Hanoi graphs H3

3 and H2 4

Definitions Main result

Sierpi´ nski graphs

Sierpi´ nski graph Sn

p

• V (Sn

p ) = [p]n

• two vertices sn . . . s1 and rn . . . r1 are adjacent if there exists

an index δ, such that

(i) sℓ = rℓ, for ℓ = n, . . . , δ + 1; (ii) sδ = rδ; and (iii) sℓ = rδ and rℓ = sδ for ℓ = 1, . . . , δ − 1.

Equivalently: E(S1

p) =

{i, j} | i = j ∈ {0, 1 . . . , p − 1}

• E(Sn

p ) =

{is, ir} | i = 0, 1, . . . , p − 1 , {s, r} ∈ E(Sn−1

p

) ∪ {ijn−1, jin−1} | i = j ∈ {0, 1 . . . , p − 1}

• (n ≥ 2)

Definitions Main result

Sierpi´ nski graphs

000 001 002 010 011 012 020 021 022 100 101 102 110 111 112 120 121 122 200 201 202 210 211 212 220 221 222 02 01 03 00 13 10 12 11 20 23 21 22 30 31 32 33

Figure: The Sierpi´ nski graphs S3

3 and S2 4

Definitions Main result

Notations

ii . . . i = in – extreme vertex in Sn

p or perfect vertex in Hn p

– p extreme vertices in Sn

p

– degree of any extreme vertex is p − 1, all other vertices have degree p – p perfect vertices in Hn

p

– degree of any perfect vertex is p − 1, all other vertices have degree ≥ 2p − 3 (for n ≥ 2)

sn . . . sr+1Sr

p = {sn . . . sr+1s | s ∈ Sr p} ≃ Sr p

sn . . . sr+1Hr

p = {sn . . . sr+1s | s ∈ Hr p} ≃ Hr p

Definitions Main result

Theorem

Theorem Let p, n ∈ N. Then Sn

p can be embedded isomorphically into Hn p if

and only if p is odd or n = 1.

• n = 1: H1

p = S1 p = Kp

• p odd:

– Hn

1 = Sn 1 = K1

– For p ≥ 3 define for each k ∈ [p]0 πk(i) = 1 2

• k(p + 1) − i(p − 1)
• mod p ... permutation of [p]0 ,

πn

k(sn . . . s1) = πk(sn) . . . πk(s1) ... permutation of [p]n 0 .

The embedding ιn+1 : V (Sn+1

p

) → V (Hn+1

p

) is defined by ιn+1(ks) = kπn

k (ιn(s)) .

Definitions Main result

Proof

00 01 02 03 04 10 11 12 13 14 20 21 22 23 24 30 31 32 33 34 40 41 42 43 44 00 03 01 04 02 13 11 14 12 10 21 24 22 20 23 34 32 30 33 31 42 40 43 41 44

Figure: Isomorphic embedding ι2 from S2

5 into H2 5

Definitions Main result

Proof - p even

Theorem Let p, n ∈ N. Then Sn

p can be embedded isomorphically into Hn p if

and only if p is odd or n = 1.

• p even:

– Hn

2 ... n − 1 disjoint K2

Sn

2 = P2n ... path on 2n vertices

– p ≥ 4 Lemma Every complete subgraph of Hn

p, p, n ∈ N, is induced by edges

corresponding to moves of one and the same disc. In particular, ω(Hn

p ) = p and the only p-cliques of Hn p are of the form

sn . . . s2H1

p.

Definitions Main result

Proof - p even

Theorem Let p, n ∈ N. Then Sn

p can be embedded isomorphically into Hn 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 S2 p, but there are only

p⌊ p−1

2 ⌋ non-incident edges in H2 p

– n ≥ 3: subgraph in−2S2

p has to be mapped to some jn−2H2 p, a

contradition

Definitions Main result

Proof - p even

02 01 03 00 13 10 12 11 20 23 21 22 30 31 32 33 03 02 01 00 12 13 10 11 21 20 23 22 31 32 30 33

Figure: Graphs S2

4 and H2 4

Definitions Main result

Corollary

Hamming graph ... ... is the graph with vertex set [r1] × [r2] × . . . × [rn], where two vertices are adjacent if they differ in precisely one coordinate. ... is the Cartesian product of complete graphs Kr1Kr2 . . . Krn. K n

p = KpKp . . . Kp

Corollary Let p be odd. Then for any n, Sn

p is a spanning subgraph of the

Hamming graph K n

p .