HC IN (2, 4k, 3)-CAYLEY GRAPHS

Transcription

1 HAMILTON CYCLES IN (2, 4k, 3)-CAYLEY GRAPHS University of Primorska November, 2008 Joint work with Henry Glover and Dragan Marušič

2 Definitions An automorphism of a graph X = (V, E) is an isomorphism of X with itself. Thus each automorphism α of X is a permutation of the vertex set V which preserves adjacency. An s-arc in a graph X is an ordered (s + 1)-tuple (v 0, v 1,..., v s 1, v s ) of vertices of X such that v i 1 is adjacent to v i for 1 i < s, and also v i 1 v i+1 for 1 i < s.

3 Different types of transitivity A graph is vertex-transitive if its automorphism group acts transitively on vertices. A graph is edge-transitive if its automorphism group acts transitively on edges. A graph is arc-transitive (also called symmetric) if its automorphism group acts transitively on arcs. An arc-transitive graph X is said to be s-regular if for any two s-arcs in X, there is a unique automorphism of X mapping one to the other.

4 Cayley graphs Definition Given a group G and a subset S of G \ {1}, S = S 1, the Cayley graph Cay(G, S) has vertex set G and edges of the form {g, gs} for all g G and s S. The group G acts on itself by the left multiplication. This action may be viewed as the action of G on its Cayley graph. A graph is a Cayley graph of a group G if and only if its automorphism group contains a regular subgroup isomorphic to G.

5 Cubic Cayley graphs If Cay(G, S) is a cubic Cayley graph then S = 3, and either S = {a, b, c a 2 = b 2 = c 2 = 1}, or S = {a, x, x 1 a 2 = x s = 1} where s 3.

6 (2, s, t)-cayley graphs Definition Let G = a, x a 2 = x s = (ax) t = 1,... be a group. Then the Cayley graph X = Cay(G, S) of G with respect to the generating set S = {a, x, x 1 } is called (2, s, t)-cayley graph. A (2, s, t)-cayley graph is a Cayley graph of a quotient group of the triangle group T (2, s, t). A (2, s, 3)-Cayley graph is a Cayley graph of a quotient group of the modular group a, x a 2 = (ax) 3 = 1. The modular group is the group of linear Möbius transformations of the upper half of the complex plane. The modular group can be shown to be generated by the two transformations a : z 1/z and x : z z + 1.

7 Hamiltonicity of vertex - transitive graphs A Hamilton path is a spanning path. A Hamilton cycle is a spanning cycle.

8 Lovász question Lovász, 1969 Does every connected vertex-transitive graph have a Hamilton path?

9 Hamiltonicity of vertex - transitive graphs All known VTG have Hamilton path. Only 4 CVTG (having at least three vertices) not having a Hamilton cycle are known to exist. None of them is a Cayley graph. Folklore conjecture Every connected Cayley graph contains a Hamilton cycle.

10 Hamiltonicity of (2, s, 3)-Cayley graphs Glover, Marušič, JEMS, 2007 Let s 3 be an integer and let X = Cay(G, S) be a (2, s, 3)-Cayley graph of a group G. Then X has a Hamilton cycle when G is congruent to 2 modulo 4, and a cycle of length G 2, and also a Hamilton path, when G is congruent to 0 modulo 4.

11 (2, s, 3)-Cayley graphs of order 0 (mod 4) For a (2, s, 3)-Cayley graph of order 0 (mod 4) three cases can occur: s 0 (mod 4). s 2 (mod 4). s odd.

12 Hamiltonicity of (2, s, 3)-Cayley graphs, s 0 (mod 4) Glover, KK, Marušič Let s 0 (mod 4) 4 be an integer. Then a (2, s, 3)-Cayley graph X has a Hamilton cycle. Essential ingredients in the proof Glover-Marušič method (Glover, Marušič, JEMS, 2007). Classification of cubic ATG of girth 6 (KK, Marušič, JCTB 2008). Results on cubic ATG admitting a 1-regular subgroup.

13 Essential ingredients in the proof A (2, s, 3)-Cayley graph X can be embedded in the closed orientable surface of genus 1 + (s 6) G /12s with faces G /s disjoint s-gons and G /3 hexagons.

14 Soccer ball (2, 5, 3)-Cayley graph of A 5 = a, x a 2 = x 5 = (ax) 3 = 1.

15 The hexagon graph Hex(X ) G = a, x a 2 = x s = (ax) 3 = 1,...

16 The hexagon graph Hex(X ) G = a, x a 2 = x s = (ax) 3 = 1,... Hex(X ) is isomorphic to the orbital graph of the left action of G on the set of left cosets H of the subgroup H = ax, arising from the suborbit {ah, x 1 H, ax 2 H} of length 3. More precisely, the graph has vertex set H, with adjacency defined as follows: yh yah(= yxh), yx 1 H, yax 2 H(= yaxah) y G.

17 The hexagon graph Hex(X ) G = a, x a 2 = x s = (ax) 3 = 1,... Hex(X ) is isomorphic to the orbital graph of the left action of G on the set of left cosets H of the subgroup H = ax, arising from the suborbit {ah, x 1 H, ax 2 H} of length 3. More precisely, the graph has vertex set H, with adjacency defined as follows: yh yah(= yxh), yx 1 H, yax 2 H(= yaxah) y G. G acts 1-regularly on Hex(X ), and so it is a cubic symmetric graph.

18 Types of cubic symetric graphs The 17 families of finite cubic symmetric graphs (Conder, Nedela, 2006): s Type s Type , , , 4 1, 4 2, 5 2 1, 2 1, , 4 2, , 2 2, , , , ,

19 Soccer ball (2, 5, 3)-Cayley graph of A 5 = a, x a 2 = x 5 = (ax) 3 = 1.

20 Cyclically stable subsets

21 Cyclically k-edge-connected graphs Cycle-separating subset A subset F E(X ) of edges of X is said to be cycle-separating if X F is disconnected and at least two of its components contain cycles. Cyclically k-edge-connected graphs A graph X is cyclically k-edge-connected, if no set of fewer than k edges is cycle-separating in X. Nedela, Škoviera, 1995 Cyclic edge-connectivity of a vertex-transitive graph equals its girth.

22 Cubic not c-4-c graph

23 Payan, Sakarovitch, 1975 Payan, Sakarovitch, 1975 Let X be a cyclically 4-edge-connected cubic graph of order n, and let S be a maximum cyclically stable subset of V (X ). Then S = (3n 2)/2 and more precisely, the following hold. If n 2 (mod 4) then S = (3n 2)/4, and X [S] is a tree and V (X ) \ S is an independent set of vertices; If n 0 (mod 4) then S = (3n 4)/4, and either X [S] is a tree and V (X ) \ S induces a graph with a single edge, or X [S] has two components and V (X ) \ S is an independent set of vertices.

24 Modification process in (2, 4k, 3) case We consider the graph of hexagons HexX. The graph of hexagons HexX is modified so as to get a graph ModX with 2(mod 4) vertices and cyclically 4-edge-connected, so to be able to get by [Payan, Sakarovitch, 1975] a tree whose complement is an independent set of vertices giving rise to a Hamilton cycle in X.

25 Modification process in (2, 4k, 3) case The local structure

26 Modification process in (2, 4k, 3) case Graph of hexagons Hex(X )

27 Modification process in (2, 4k, 3) case Graph of hexagons Hex(X ) Hex(X ) is of order G /3.

28 Modification process in (2, 4k, 3) case

29 Modification process in (2, 4k, 3) case

30 Modification process in (2, 4k, 3) case

31 Modification process in (2, 4k, 3) case Modified graph ModX is of order V (HexX ) (3s 6). Since V (HexX ) 0(mod 4) we have that ModX is of order 2(mod 4).

32 Modification process in (2, 4k, 3) case HexX is cubic symmetric graph. Lemma If Hex(X ) is cyclically c-edge-connected for c 5 then it is Θ 2, K 4, K 3,3, Q 3 or GP(10, 2). If Hex(X ) cyclically 6-edge-connected then it is an Ik n (r)-path or a suitable cover of the cube or some Ik n(r)-path. If Hex(X ) is cyclically 7-edge-connected then s = 7. In all other cases Hex(X ) is cyclically 8-edge-connected.

33 Modification process in (2, 4k, 3) case Using this lemma we show ModX is cyclically 4-edge-connected. If cyclical edge-connectivity c of HexX is at most 5 then HexX is Θ 2 or Q 3 ; If c = 6 then it is first proved that HexX is some Ik n (r)-path and then shown that the modification process does not reduce c by more than 2. The case c = 7 is impossible; If c is at least 8, then it is shown that the modification process does not reduce c by more than 4. Applying [Payan, Sakarovitch] to ModX gives a Hamilton cycle in X.

34 I n k (r)-path, k 2 Using Frucht s notation.

35 Precise description of CSG6 A cubic symmetric graph of girth 6 different from the Moebius-Kantor graph is 2-regular if and only if it is an I n k (r)-path where k = n/2 or k = n/6, r = n/2 if k is odd and r = n/2 + 1 if k is even. A cubic symmetric graph of girth 6 is 1-regular if and only if it is a suitable cover of the cube or some I n k (r)-path.

36 Consistent cycles Let X be a graph and G Aut(X ). A walk α = (v 0,..., v r ) in X is called G-consistent if there exists g G such that v g i = v i+1 for i {0, 1,..., r 1}. The automorphism g is called a shunt automorphism for α. (Conway,1971, Biggs 1978) Let G be a group of automorphisms of a d-valent graph X (d 2). Assume that G is arc-transitive. Let Ω be the set of all G-consistent cycles in X. Then G has exactly d 1 orbits in its action on Ω. Further results on this topic by Miklavič, Potočnik, Willson, In cubic case: an arc-transitive group G has 2 orbits of consistent cycles.

37 Consistent cycles in cubic symmetric graphs of girth 6 In a 2-regular CSG6 consistent cycles are of length 6 and n > 6. The length of consistent cycles in a 1-regular CSG6 is 6.

38 Example Graph Order Group s Girth Consistent cycles F 032A , 6 Graph Order Group s Girth Consistent cycles F 050A , 6

39 Example

40 Thanks!