The Elusive Hamilton Problem for Cayley Graphs|an Overviewgoddard/MINI/2013/Westlund.pdf · The Elusive Hamilton Problem for Cayley Graphs|an Overview 28th Clemson Mini-Conference

The Elusive Hamilton Problem for Cayley Graphs—anOverview

28th Clemson Mini-Conference on Discrete Mathematics and Algorithms

Erik E. Westlund

Department of Mathematics and StatisticsKennesaw State University

October 3, 2013

Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 1 / 43

Preliminaries

Definition

G finite group; S ⊆ G − {e}, S = S−1, the Cayley graph of G with connection set S

X = Cay(G ; S)

V (X ) = G

{x , y} ∈ E(X )⇔ x−1y ∈ S , i.e., E(X ) = {{x , xs} : s ∈ S}

Cay(Z12; {2, 3, 4}?) Cay(D4; {a, b}?) Cay(Z42; S?)


Preliminaries

Definition

G finite group; S ⊆ G − {e}, S = S−1, the Cayley graph of G with connection set S

X = Cay(G ; S)

V (X ) = G

{x , y} ∈ E(X )⇔ x−1y ∈ S , i.e., E(X ) = {{x , xs} : s ∈ S}

Remark

X connected ⇔ 〈S〉 = G

e = {x , y} is a t-edge if x−1y = t ∈ S

subgraph generated by t consists of all t-edges

involution generates a 1-factor.

non-involution generates a 2-factor (cycle ↔ coset a〈t〉).

Cayley graphs are vertex-transitive (VT). Identical “local neighborhoods.” Tons ofsymmetry.


Terminology

Definition

S is inverse-free if for all x ∈ S , either |x | = 2 or −x /∈ S .

S is involution-free if |x | 6= 2 for all x ∈ S .

S is inverse-closed if whenever x ∈ S then −x ∈ S.

S? is the inverse-closure of S : the smallest superset of S that is inverse-closed.

Remark

If S = {s1, . . . , sk} is an inverse- and involution-free generating set for A, then X = Cay(A; S?)is connected and 2k-regular.


44 years ago....

In 1969, Lovasz posed this problem:

“Let us construct a finite, connected undirected graph which is [vertex-transitive] and has nosimple path containing all vertices.”

Conjecture (Lovasz et al. c. 1969)

Every finite, connected, vertex-transitive graph has a Hamilton path.

MASSIVE AMOUNT OF RESEARCH & RICH LITERATURE ⇒ No known counterexamples!

Only FOUR finite, connected nontrivial VT graphs are known that do not possess a Hamiltoncycle.

NONE OF THE FOUR ARE CAYLEY GRAPHS.

Conjecture (Many people....)

Every connected, Cayley graph on a finite group has a Hamilton cycle, except for K2.


Done for Abelian groups!

Theorem (Folklore)

Every nontrivial, connected Cayley graph of a finite Abelian group has HC.

Approaches.

1 Method 1: Abelian, transitive subgroup of automorphisms [Lovasz 1979]

2 Method 2: Chen-Quimpo Theorem [Chen-Quimpo 1981]

3 Method 3: induction using M-sequences [Marusic 1983]

4 Method 4: induction on |S | [Witte ?]

Cayley graphs of Abelian groups flourish with Hamilton cycles.

Assumptions hereafter: A is a finite Abelian group. S ⊆ A. All notation is additive.


A conjecture of Alspach

Conjecture (Alspach 1984)

If X = Cay(A; S) is connected and 2k-regular, then X is decomposable into k Hamilton cycles.

Theorem

Known results (k = 1 is trivial):

k = 2 (Bermond et al. 1989, Fan et al. 1996)

k = 3 and A ∼= Z2n+1 or A ∼= Z2n with a generator in S (Dean 2006, 2007)

k = 3 and |A| = 2n + 1 (Fan et al. 1996, Kreher, Liu, W. 2009)

|A| = 2n + 1, S is minimal (Liu 1996)

|A| = 2n ≥ 4, S is strongly minimal (Liu 2003)

|A| ∈ {p, 2p, pq, p2}, odd primes (Liu 1993, Li et al. 2000)

X cartesian product of cycles and 4-regular Cayley graphs (Alspach et al. 1992)

X is a Payley graph (Alspach et al. 2012)

numerous others (e.g., Alspach et al. 2013)


Terminology

Definition

Let S ⊂ A. Fix an a ∈ S .

S is minimal if x /∈ 〈S − {x}〉 for all x ∈ S .

S is strongly minimal if 2x /∈ 〈S − {x}〉 for all x ∈ S .

S is strongly a-minimal if 2x /∈ 〈a〉 for all x ∈ S − {a}.

Strongly-minimal ⇒ minimal, involution-free, strongly a-minimal ∀a ∈ S.

Strongly a-minimal does not imply strongly-minimal or minimal.

Strongly a-minimal for non-involution a implies involution-free.

Even strongly a-minimal for ALL a ∈ S does not imply strongly-minimal or minimal.

S = {(1, 0), (0, 1), (3, 1)} ⊂ Z4 ⊕ Z4 is strongly a-minimal for all a ∈ S .


Quotients of Cayley Graphs

Definition

Let S = {s1, . . . , sk} generate A and S = {s1, . . . , sk−1}, where si = si + 〈sk 〉 for all i . The

Cayley graph, X = Cay(A; S) where A = A/〈sk 〉 is a quotient graph of X = Cay(A;S).

6-regular Cayley graph and one of its cubic quotient graphs (not 4-regular because S is notstrongly 4-minimal).


Lifting Edge Sets

Definition

Let S = {s1, . . . , sk} be inverse-free.

Lift of the si -edge {a, b} ∈ E(X ) is

LX {a, b} := {{x , y} : a = x , b = y , y − x = si}.

|LX {a, b}| = |sk |.Lift of the subgraph F (of X ) is the subgraph F (of X ) induced on⋃

{a,b}∈E(F )

LX {a, b}.

Edge-disjoint subgraphs in X lift to edge-disjoint subgraphs in X .


Lifting Example

X = Cay(Z12; {2, 3, 4}?) and X = Cay(Z12/〈4〉; {2, 3}?)

LX {3, 2} = {{x , y} : x = 3, y = 2, y − x = 3}

3 2+3

7 10

3 6+3

11 2

3 + 〈4〉 2 + 〈4〉

LX {3, 1} = {{x , y} : x = 3, y = 1, y − x = 2}

3 1+2

7 9

3 5+2

11 1

3 + 〈4〉 1 + 〈4〉


r -pseudo-cartesian products

Definition

For a fixed r ∈ Zm, the graph An ×r Bm is the r -pseudo-cartesian product of An = a1, . . . , anand Bm = b1, . . . , bm:

V = {(ai , bj ) : 1 ≤ i ≤ n, 1 ≤ j ≤ m}E = EH ∪ EV ,

EH = {{(ai , bj ), (ai+1, bj )}, {(an, bj ), (a1, bj+r )} : 1 ≤ i < n, 1 ≤ j ≤ m}EV = {{(ai , bj ), (ai , bj+1)}, {(ai , b1), (ai , bm)} : 1 ≤ i ≤ n, 1 ≤ j < m}

A10 ×2 B8


r -pseudo-cartesian products

Fact

EH is a 2-factor H, of An ×r Bm consisting of t = gcd(r ,m) cycles of length nm/t.

bi -row and bj -row on same cycle ⇔ j ≡ i (mod t)

Any t consecutive rows on t different cycles.

An ×r Bm∼= Cay(A; {s1, s2}?), where ns1 = rs2.

An ×r Bm is HD for all n,m ≥ 3. [Fan et al. (1996)]


Lifting Hamilton Cycles I

Theorem (Liu 1994)

If S = {s1, . . . , sk} is an inverse- and involution-free generating set for A, and F is the 2-factor

generated by sk , then any Hamilton cycle of Cay(A/〈sk 〉;S?

) lifts to a 2-factor, H, ofCay(A; S?), such that H ∪ F ∼= An ×r Bm.

X = Cay(Z12; {2, 3, 4}?) and X = Cay(Z12/〈4〉; {2, 3}?)


D3(m, n)-graphs and Cayley Graphs

If Hamilton cycle lifts to a 2-factor, then Hamilton decomposition lifts to a 2-factorization.

Definition (Liu 1994)

G is a Dk(m, n)-graph if, for m, n,≥ 3:

1. V (G) = {(ai , bj ) : 1 ≤ i ≤ n and 1 ≤ j ≤ m}

2. E(G) = F ∪ H1 ∪ · · · ∪ Hk−1, and for t ∈ {1, . . . , k − 1},

F ∪ Ht∼= A

(t)n ×rt Bm

where A(t)n = at

σt (1)atσt (2)

· · · atσt (n)

atσt (1)

and Bm = b1b2 · · · bmb1, for some σt ∈ Sym(n), and

(ati , bj ) = (ai , bj+hi,t ) for some 0 ≤ hi,t < m.

Color edges in H1 blue, edges in H2 black, and edges in F red. Dk (m, n)-graphs are idealplatforms to perform edge color-switching.


Cay(Z6 ⊕ Z8; {(3, 1), (1, 2), (5, 7)}?) is a D3(8, 6)-graph

S is strongly (3, 1)-minimal, X = Cay(A/〈(3, 1)〉; {(1, 2), (5, 7)}?) is HD:

H1 = (0, 0), (1, 2), (5, 2), (4, 0), (3, 6), (5, 7), (0, 0)

H2 = (0, 0), (5, 2), (3, 6), (1, 2), (5, 7), (4, 0), (0, 0).

a1 a2 a3 a4 a5 a6

b8

b7

b6

b5

b4

b3

b2

b1

3,7 4,1 5,2 4,0 3,6 2,4

0,6 1,0 2,1 1,7 0,5 5,3

3,5 4,7 5,0 4,6 3,4 2,2

0,4 1,6 2,7 1,5 0,3 5,1

3,3 4,5 5,6 4,4 3,2 2,0

0,2 1,4 2,5 1,3 0,1 5,7

3,1 4,3 5,4 4,2 3,0 2,6

0,0 1,2 2,3 1,1 0,7 5,5

a21 a2

3 a25 a2

2 a26 a2

4

b8

b7

b6

b5

b4

b3

b2

b1

3,7 2,5 3,6 4,7 5,1 4,0

0,6 5,4 0,5 1,6 2,0 1,7

3,5 2,3 3,4 4,5 5,7 4,6

0,4 5,2 0,3 1,4 2,6 1,5

3,3 2,1 3,2 4,3 5,5 4,4

0,2 5,0 0,1 1,2 2,4 1,3

3,1 2,7 3,0 4,1 5,3 4,2

0,0 5,6 0,7 1,0 2,2 1,1

H1 ∪ F ∼= A(1)6 ×6 B8 H2 ∪ F ∼= A

(2)6 ×0 B8

{F : R6;H1 : BL2;H2 : BK8}


Using quotient graphs

Theorem (Liu 1996)

Let X = Cay(A; {s1, . . . , sk}?) and B = 〈sk 〉. If X = Cay(A/B; {s1, . . . , sk−1}?) can be

decomposed into k − 1 Hamilton cycles, Hi , then X is a Dk (m, n)-graph, with m = |B|,n = |A : B|, where F is the 2-factor generated by sk and Hi is the lift of Hi , for i = 1, . . . , k − 1.

Focus on S = {s1, s2, s3} involution-free, inverse-free, strongly s3-minimal:

X = Cay(A; S?) is 6-regular ⇒ X is 4-regular ⇒ X is D3(m, n)-graph


Decomposing D3(m, n)-graphs

Theorem (Bermond et al., Dean, Fan et al., Li et al., Liu, Westlund et al.)

A D3(m, n)-graph is HD if:

(a) m, n ≥ 3 and t1 = 1 or t2 = 1.

(b) m ≥ 3 is odd, and n ∈ {3, 5, 7} or n ≥ 9.

(c) m ≥ 4 is even, and n ≥ 9, and t1 and t2 are odd.

(d) m ≥ 4 is even, and n ≥ 10 is even, t1 is even, t2 is odd.

(e) m ≥ 6 is even, and n ≥ 14 is even, t1 and t2 are even.

Remark (Open Cases)

Low-Order m ≥ 3; 3 ≤ n ≤ 8; mn is even.

High-Order m ≥ 4 is even; n ≥ 9 is odd; t1 is even, t2 is odd.

High-Order m ≥ 4 is even; 9 ≤ n ≤ 12 or n ≥ 13 is odd; t1 and t2 both even.


Color-switching fundamentals

In An ×r Bm, color EH blue and color EV red.

Definition

An {ai , ai+1, bj , bj+1}-color switch is an operation that interchanges the color of the edges

{{(ai , bj ), (ai+1, bj )}, {(ai , bj+1), (ai+1, bj+1)}}

with the color of the edges

{{(ai , bj ), (ai , bj+1)}, {(ai+1, bj ), (ai+1, bj+1)}}.

A color-switching configuration, abbreviated CSC, is a set of color-switches that are pairwiseedge-disjoint.



Suppose C1 and C2 are vertex-disjoint cycles in a graph X , and {xi , yi} ∈ E(Ci ) for i ∈ {1, 2}. IfC = x1y1x2y2 is a cycle of length four in X , and the edges {y1, x2} and {y2, x1} are not inE(C1) ∪ E(C2), then the subgraph of X whose edge set is the symmetric difference

(E(C1) ∪ E(C2))⊕ E(C)

is a single cycle.



{ai , ai+d , bj}-LAHS {ai , ai+d , bj}-RAHS

{ai , bj , bj+d}-LAVS {ai , bj , bj+d}-RAVS


Example: Cay(Z42; {6, 3, 7}?)

{F : R7,H1 : BL3,H2 : BK6}

X = Cay(Z42, {6, 3, 7}?), X = Cay(Z42/〈7〉, {6, 3}?) ∼= Cay(Z7, {1, 2}?). X is a D3(6, 7)-graph

with F ∪ H1∼= A

(1)7 ×3 B6 and F ∪ H2

∼= A(2)7 ×0 B6. (vertical red edges shown twice.)


Example: Cay(Z42; {6, 3, 7}?)

{S1(F) : R5,S1(H1) : BL

1,H2 : BK6}

S1 = {a2, b1, b3}-RAVS


Example: Cay(Z42; {6, 3, 7}?)

{S1(F) : R5,S1(H1) : BL

1,H2 : BK6}

Orient the blue Hamilton cycle and search for a “good” 4-cycle.


Example: Cay(Z42; {6, 3, 7}?)

{S2S1(F) : R4,S2(S1(H1)) : BL

1,H2 : BK6}

S2 = {a3, a4, b3, b4}-CS


Example: Cay(Z42; {6, 3, 7}?)

{S3S2S1(F) : R2,S2S1(H1) : BL

1,S3(H2) : BK2}

S3 = {a25, b1, b5}-RAVS

Case Analysis: search for a properly 2-edge-colored red/black 4-cycle to make the final switch.


Example: Cay(Z42; {6, 3, 7}?)

{S4S3S2S1(F) : R1,S2S1(H1) : BL

1,S4S3(H2) : BK1}

S4 = {a24, a

26, b5, b6}-CS

Three monochromatic Hamilton cycles. (vertical red edges shown twice.)


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)

If S is strongly s3-minimal, |s3| = 2k ≥ 6, and |A : 〈s3〉| = 2n′ + 1 ≥ 9, then X is HD.

Partial proof sketch. Let A = A/〈s3〉 = {a1, a2, . . . , an}, n ≥ 9

1 X is D3(m, n)-graph and assume that s1 6= ±s2.

Example: X = Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?) has strongly (1, 0)-minimal S .

X = Cay((Z8 ⊕ Z11)/〈(1, 0)〉; {(4, 1), (4, 2)}?) is 4-regular circulant of order 11.

H1 = (4, 3), (0, 4), (4, 5), (0, 6), (4, 7), (0, 8), (4, 9), (0, 10), (1, 0), (4, 1), (0, 2);

H2 = (0, 6), (0, 8), (0, 10), (4, 1), (4, 3), (4, 5), (4, 7), (4, 9), (1, 0), (0, 2), (0, 4).

H1 and H2 lift to H1 and H2, each having t1 = t2 = 4 cycles.

H1 ∪ F ∼= A(1)11 ×4 B8 and H2 ∪ F ∼= A

(2)11 ×4 B8


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)


Partial proof sketch. Let A = A/〈s3〉 = {a1, a2, . . . , an}, n ≥ 9

1 X is D3(m, n)-graph and assume that s1 6= ±s2.

Example:

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

a27 a2

9 a211 a2

2 a24 a2

6 a28 a2

10 a21 a2

3 a25

b8

b7

b6

b5

b4

b3

b2

b1

{H1 : BL4,F : R11,H2 : BK4}

H1 ∪ F ∼= A(1)11 ×4 B8 and H2 ∪ F ∼= A

(2)11 ×4 B8


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)


Partial proof sketch.

2 X has a 6-path, P = aσ1(1) aσ1(2) aσ1(3) ap aσ2(2) aσ2(3), where {σ2(2), σ2(3)} = {q, n}, and4 ≤ p < q < n − 1, and p = σ1(4) = σ2(1).

a1 a2 a3 a4 a5 a6 a7

apa8 a9

aqa10 a11

an

b8

b7

b6

b5

b4

b3

b2

b1

a1 a2 a3 a4 a5 a6 a7

apa8 a9

aqa10 a11

an

b8

b7

b6

b5

b4

b3

b2

b1

a27 a

29 a

211 a2

2 a24 a2

6 a28 a2

10 a21 a2

3 a25

b8

b7

b6

b5

b4

b3

b2

b1

{H1 : BL4,F : R11,H2 : BK4}

P = (4, 3), (0, 4), (4, 5), (0, 6), (0, 8), (0, 10) = a4 a5 a6 a7 a9 a11


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)



3 S1 = {{a2, b1, bt1−1}-RAVS, {a3, a4, bt1−1, bt1}-CS} ⇒ blue Hamilton cycle, red cycle onai -columns, 1 ≤ i ≤ 4.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

a27 a2

9 a211 a2

2 a24 a2

6 a28 a2

10 a21 a2

3 a25

b8

b7

b6

b5

b4

b3

b2

b1

{S1(H1) : BL1,S1(F ) : R8,H2 : BK4}S1 = {{a2, b1, b3}-RAVS, {a3, a4, b3, b4}-CS}


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)



4 S2 = {a4, an−1, bt1−1}-RAHS to create a red 2-factor with 2 cycles, preserve the blue cycle.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

�

�

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

�

�

a27 a2

9 a211 a2

2 a24 a2

6 a28 a2

10 a21 a2

3 a25

b8

b7

b6

b5

b4

b3

b2

b1

{S2S1(H1) : BL1,S2S1(F ) : R2,H2 : BK4}S2 = {a4, a10, b3}-RAHS


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)



5 Remove the switches incident with the aq-column. This preserves the blue cycle and

creates a red 2-factor with four cycles:

♦-cycle on ai -columns, 1 ≤ i ≤ q − 1.

♣-cycle on aq-column and the ♥-cycle on an-column.

♠-cycle on ai -columns, q + 1 ≤ i ≤ n − 1.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

a1

♦a2

♦a3

♦a4

♦a5

♦a6

♦a7

♦a8

♦a9

♣a10

♠a11

♥

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

a27 a2

9 a211 a2

2 a24 a2

6 a28 a2

10 a21 a2

3 a25

b8

b7

b6

b5

b4

b3

b2

b1

{S2S1(H1) : BL1,S2S1(F ) : R4,H2 : BK4}


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)



6 Apply S3 = {{a2σ2(2)

, b1+`, bt2−1+`}-LAVS or -RAVS, for “nice” `, producing:

red 2-factor with 2 cycles: ♠-cycle and F-cycle.

black 2-factor with 2 cycles: (1) containing all edges in bj+`-rows, 1 ≤ j ≤ t2 − 1, (2)

containing edges in bkt2+`-rows.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

a1

Fa2

Fa3

Fa4

Fa5

Fa6

Fa7

Fa8

Fa9

Fa10

♠a11

F

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

a27

F

a29

F

a211

F

a22

F

a24

F

a26

F

a28

F

a210

♠a2

1

F

a23

F

a25

F

b8.

b7◦

b6◦

b5◦

b4.

b3◦

b2◦

b1◦

�

�

{S2S1(H1) : BL1,S3S2S1(F ) : R2,S3(H2) : BK2}S3 = {a2

9, b1, b3}-LAVS


Cay(Z8 ⊕ Z11; {(4, 1), (4, 2), (1, 0)}?)

Lemma (W.)



7 z = min3≤i<n{a2σ2(i)

-column is on F-cycle and a2σ2(i+1)

-column is on ♠-cycle}.

8 Technical case analysis ⇒ S4 = {a2σ2(z)

, a2σ2(z+1)

, bk , bk+1}-CS ⇒ HD.

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

a1 a2 a3 a4 a5 a6 a7 a8 a9 a10 a11

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

�

�

�

a27 a2

9 a211 a2

2 a24 a2

6 a28

z

a210

z + 1

a21 a2

3 a25

b8

b7

b6

b5

b4

b3

b2

b1

�

�

�

�

{S2S1(H1) : BL1,S4S3S2S1(F ) : R1,S4S3(H2) : BK1}S4 = {{a2

8, a210, b8, b1}-CS}


Basis for Main Result

Theorem (Fan et al. 1996)

If X = Cay(A; {s1, s2, s3}?) is connected, 6-regular, and has a strongly s3-minimal connectionset, where |s3| is odd, and |A : 〈s3〉| ≥ 9, then X is HD.

Corollary (Fan et al. 1996)

If X = Cay(A; {s1, s2, s3}?) is connected, 6-regular, has odd order, and |s1| ≥ |s2| > |s3|, then Xis HD.


New Results for 6-Regular Graphs

Theorem (W.)

If X = Cay(A; {s1, s2, s3}?) is connected, 6-regular, and has a strongly s3-minimal connectionset, then X is HD whenever

1 |A : 〈s3〉| ≥ 4; or

2 |A : 〈s3〉| = 3 and 〈s1〉 and 〈s2〉 have even index.

Hypotheses forbid |A : 〈s3〉| = 2: we have partial results.Dean resolves case when |A : 〈s3〉| = 1.

Corollary (W.)

If X = Cay(A; {s1, s2, s3}?) is connected, 6-regular, has even order, and |s1| ≥ |s2| > 2|s3|, thenX is HD.

Even when S is very non-minimal:

Corollary (W.)

If X has even order, A = 〈s1, s3〉 = 〈s2, s3〉, and |A : 〈s3〉| ≥ 4, then X is HD.


New Results for 6-Regular Graphs

Theorem (W. 2012)

If A = 〈s1, s2〉 and |A : 〈s3〉| = 2, then X is HD.

E.g., if A has order 2n, where n is square-free, then has cyclic subgroups of index two.

Corollary

Circ(2n; {a, b, c}?) is HD ifgcd(2n, a, b) · gcd(2n, c) = 2.

E.g., Cay(Z2c ; {2a + 1, b, 2}) is HD.

Corollary

If s1 generates a 2-factor consisting of two cycles in X , and s2 generates a Hamilton cycle in X ,then X is HD.


Open Problems

1 Find HD if |A : 〈s3〉| = 3 and at least one of 〈s1〉 and 〈s2〉 has odd index.

2 Find HD if S is not strongly a-minimal for any a ∈ S.

3 Find HD if S is not involution-free.


Thank you.


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Documents

The Elusive Hamilton Problem for Cayley Graphs|an Overviewgoddard/MINI/2013/Westlund.pdf · The Elusive Hamilton Problem for Cayley Graphs|an Overview 28th Clemson Mini-Conference