The Elusive Hamilton Problem for Cayley Graphs—an Overview 28th Clemson Mini-Conference on Discrete Mathematics and Algorithms Erik E. Westlund Department of Mathematics and Statistics Kennesaw State University October 3, 2013 Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 1 / 43
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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
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 16 / 43
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
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 17 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 18 / 43
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
A color-switching configuration, abbreviated CSC, is a set of color-switches that are pairwiseedge-disjoint.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 19 / 43
Color-switching fundamentals
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 20 / 43
Color-switching fundamentals
{ai , ai+d , bj}-LAHS {ai , ai+d , bj}-RAHS
{ai , bj , bj+d}-LAVS {ai , bj , bj+d}-RAVS
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 21 / 43
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.)
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 22 / 43
Example: Cay(Z42; {6, 3, 7}?)
{S1(F) : R5,S1(H1) : BL
1,H2 : BK6}
S1 = {a2, b1, b3}-RAVS
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 23 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 24 / 43
Example: Cay(Z42; {6, 3, 7}?)
{S2S1(F) : R4,S2(S1(H1)) : BL
1,H2 : BK6}
S2 = {a3, a4, b3, b4}-CS
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 25 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 26 / 43
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.)
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 27 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 35 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 36 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 37 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 38 / 43
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.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 39 / 43
Thank you.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 40 / 43
References I
Combinatorial structures and their applications, Proceedings of the Calgary InternationalConference on Combinatorial Structures and their Applications held at the University ofCalgary, Calgary, Alberta, Canada, June, vol. 1969, Gordon and Breach Science Publishers,New York, 1970.
B. Alspach, Research problem 59, Discrete Math. 50 (1984), 115.
B. Alspach, D. Bryant, and D. Dyer, Paley graphs have Hamilton decompositions, DiscreteMath. 312 (2012), no. 1, 113–118.
B. Alspach, C. Caliskan, and D. L. Kreher, Orthogonal projection and liftings ofHamilton-decomposable Cayley graphs on abelian groups, Discrete Math. 313 (2013),no. 13, 1475–1489.
B. Alspach, K. Heinrich, and G. Z. Liu, Orthogonal factorizations of graphs, Contemporarydesign theory, Wiley-Intersci. Ser. Discrete Math. Optim., Wiley, New York, 1992,pp. 13–40.
J.-C. Bermond, O. Favaron, and M. Maheo, Hamiltonian decomposition of Cayley graphsof degree 4, J. Combin. Theory Ser. B 46 (1989), no. 2, 142–153.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 41 / 43
References II
J. A. Bondy, Hamilton cycles in graphs and digraphs, Proceedings of the NinthSoutheastern Conference on Combinatorics, Graph Theory, and Computing (FloridaAtlantic Univ., Boca Raton, Fla., 1978) (Winnipeg, Man.), Congress. Numer., XXI, UtilitasMath., 1978, pp. 3–28. MR 527929 (80k:05074)
C. C. Chen and N. F. Quimpo, On strongly Hamiltonian abelian group graphs,Combinatorial mathematics, VIII (Geelong, 1980), Lecture Notes in Math., vol. 884,Springer, Berlin, 1981, pp. 23–34.
M. Dean, On Hamilton cycle decomposition of 6-regular circulant graphs, Graphs Combin.22 (2006), no. 3, 331–340.
, Hamilton cycle decomposition of 6-regular circulants of odd order, J. Combin.Des. 15 (2007), no. 2, 91–97.
C. Fan, D. R. Lick, and J. Liu, Pseudo-Cartesian product and Hamiltonian decompositionsof Cayley graphs on abelian groups, Discrete Math. 158 (1996), no. 1-3, 49–62.
H. Li, J. Wang, and L. Sun, Hamiltonian decomposition of Cayley graphs of orders p2 andpq, Acta Math. Appl. Sinica (English Ser.) 16 (2000), no. 1, 78–86. MR 1757325
J. Liu, Hamiltonian decompositions of Cayley graphs on abelian groups, Discrete Math. 131(1994), no. 1-3, 163–171.
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References III
, Hamiltonian decompositions of Cayley graphs on abelian groups of odd order, J.Combin. Theory Ser. B 66 (1996), no. 1, 75–86.
, Hamiltonian decompositions of Cayley graphs on abelian groups of even order, J.Combin. Theory Ser. B 88 (2003), no. 2, 305–321.
D. Marusic, Hamiltonian circuits in Cayley graphs, Discrete Math. 46 (1983), no. 1, 49–54.MR 708161 (85a:05039)
G. Sabidussi, On a class of fixed-point-free graphs, Proc. Amer. Math. Soc. 9 (1958),800–804.
E. E. Westlund, Hamilton decompositions of certain 6-regular Cayley graphs on Abeliangroups with a cyclic subgroup of index two, Discrete Math. 312 (2012), 3228–3235.
, Hamilton decompositions of 6-regular Cayley graphs on even order Abelian groupswith involution-free connection sets, 2013.
E. E. Westlund, J. Liu, and D. L. Kreher, 6-regular Cayley graphs on abelian groups of oddorder are hamiltonian decomposable, Discrete Math. 309 (2009), no. 16, 5106–5110.
Erik Westlund (KSU) Elusive Hamilton Problem for Cayley Graphs October 3, 2013 43 / 43