1 Construction of Hamiltonian Cycles in Layered Cubic Planar Graphs Graphs and Combinatorics 2002 D.S. Franzblau Department of Mathematics, CUNY/College of Staten Island

2 Hamiltonian Cycles and Edge Coloring Definition Proof that Layered Graphs are Hamiltonian Reduction Lemmas Non uniform Layered Graphs Further Open Questions

3 Hamiltonian Cycles and Edge Coloring In the late nineteenth century, P. G. Tait showed that the Four Color Theorem (FCT) is equivalent to the existence of edge three-colorings for all three-connected cubic planar graphs. Tait conjectured incorrectly that every three-connected cubic planar graph has Hamiltonian cycle. Tutte proved that every four-connected planar graph is Hamiltonian. The problem of deciding whether cubic planar graphs are Hamiltonian is NP-complete. Which three-connected cubic planar graphs are Hamiltonian?

4 The main result The main result of this paper is a partial answer: a characterization of a new class of Hamiltonian cubic planar graphs. Theorem 1 –Every layered cubic planar graph has Hamiltonian cycle.

5 Motivations Thomas Hull asked for classes of convex polyhedra with cubic skeletons that can be easily edge three-colored, such as the dodecahedron. The graph determined by the skeleton of any convex polyhedron is a (three- connected) planar graph.

6 Layered cubic planar graphs

7 Definitions P(k 1,k 2,…,k n ) (k 1  2; k i  1 for i > 1) –n+1 cycles, C 0,C 1,…,C n –Each pair of successive cycles, C i, C i+1, is joined by a matching. –k 1 parallel (non-crossing) edges join cycles C 0 and C 1, creating k 1 faces. –k i edges are added incident to each face created by the matching that joins C i-2 and C i-1. (i>1)

8 The skeleton of a regular dodecahedron is isomorphic to P(5,1). n-layer cubic planar graph  n-layer graph  layered graph The boundary of a Hamiltonian polygon is a Hamiltonian cycle in G, and, conversely, the set of faces interior to any Hamiltonian cycle form a Hamiltonian polygon.

9 Hamiltonian polygon tab k 2 =5(odd) k 2 =6(even)

10 Layered cubic planar Layer i –The set of faces bounded by the cycles C i-1 and C i, along with the edges of the matching joining them Central face –The single face bounded by C 0 Outer face –The (unbounded) face bordered only by C n Hole –The faces at layer n that are not in Hamiltonian polygon.

11 Theorem 1 Every layered cubic planar graph has a Hamiltonian cycle. –Base case n=1,2 –Inductive step (n  3) To extend a Hamiltonian polygon H’ for the (n-2)- layer subgraph P’=P (k 1,k 2,…,k n-2 ) to a Hamiltonian polygon P= P(k 1,k 2,…,k n ). Five cases: k n =1 and k n-1  {1,2,3}, or k n =2 and k n-1  {1,2,}. (Lemma 1,2,3)

12 Hypothesis and observations Induction hypothesis: –H’ does not contain the outer face. Given any pair of adjacent faces in layer n-2, at least one must be in H’ Hole –H’ cannot contain a cycle –At least one face in layer n-2 must not be in H’

13 Construction strategy To construct an identical extension to H’ at each hole in such a way that the extensions can be connected. Fig. 3-5 –Three of the five cases (n,n-1)={(1,1),(1,2),(1,3),(2,1),(2,2)}

14 Construction case 1 k n =1, k n-1 =2 k n-1 =2 k n =1

15 Construction case 2,3 k n-1 =3 k n-1 =2 k n =1 k n =2

16 Observation 1 Given any two adjacent faces a and b in layer n of P’, at least one must be in H’ – to cover the vertex on cycle C n that a and b share.

17 Observation 2 More generally, if a face x in layer i-1 (or i+1) is not in the Hamiltonian polygon H’, and a and b are two faces in layer i adjacent both to x and to each other, then at least one of a and b must be in H’.

18 Observation 3 If a face x in layer i-1 (or i+1) is in the Hamiltonian polygon H’, and a and b are two faces in layer i adjacent both to x and to each other, then at most one of a and b can be in H’ – to avoid creating a three-cycle (in the dual graph of G)

19 Lemma 1 For n  3, if a Hamiltonian polygon not containing the outer face can be constructed for any n-layer graph with k n  {1,2}, then a Hamiltonian polygon not containing the outer face can be constructed for any n- layer graph.

20 Lemma 1 Face x and its leftmost adjacent face a in layer n hole k n =k+2m, k  {1,2} x  H’,b  H’ x  H’,b  H’

21 Lemma 2 For n  3, if a Hamiltonian polygon not containing the outer face can be constructed for any n-layer graph with k n =1 and k n  {1,2,3}, then a Hamiltonian polygon not containing the outer face can be constructed for any n-layer graph with k n =1.

22 Lemma 2 k n =1 and k n-1 =k+3m, k  {1,2,3} Face x and its leftmost adjacent face a in layer n-1 x  H’,b  H’,v  H’ x  H’,b  H’,v  H’

23 Lemma 3 For n  3, if a Hamiltonian polygon not containing the outer face can be constructed for any n-layer graph with k n =2 and k n  {1,2}, then a Hamiltonian polygon not containing the outer face can be constructed for any n-layer graph with k n =2.

24 Lemma 3 k n =2 and k n-1 =k+2m, k  {1,2} Face x and its leftmost adjacent face a in layer n-1 x  H’,a  H’,t  H’ x  H’,a  H’,u  H’ t is adjacent only to au is adjacent to a,b

25 Construction algorithm Build first one or two layers –The number of layers is odd, start from Fig. 2A –The number of layers is even, start from Fig. 2B Each time, add two layers, n-1 and n. –Select the appropriate one of five cases based on the value of k n (mod 2) –e.g. k n =8 and k n-1 =5  k n =2, k n-1 =1 Extend  adding new edges for each face x in layer n-2 in turn,Fig Expand  Fig. 7-8, Fig. 6

26 Construction algorithm The work performed in each stage  the number of faces in the layers added Since graph is planar  O(|V|)

27 Non uniform Layered Graphs Non uniform layered cubic planar graph –(m 1,m 2,…,m n ), m i  2 for all i A given sequence  many different graphs Example : (3,4)

28 Non uniform Layered Graphs Theorem 2 –Every non uniform layered cubic planar graph with two edges per layer, i.e., having sequence (2,2,…,2), has a Hamiltonian cycle. Theorem 3 –Every non uniform two-layer cubic planar graph has a Hamiltonian cycle. Not yet found a proof or counterexample for n  3 layers

29 Theorem 3 The number of edges from layer 1 incident to a is at least 2. Even edges between e and f odd edges between e and f

30 Theorem 3 Every face at layer 2 is incident to at most one edge in layer 1.

31 Fractional layered graph K i : fractions If m i =k 1 k 2 …k i, then m i is an integer and m i  2. E.g. (6, ½,3, 1/3)

32 Further open questions Do all such fractional layered graphs have Hamiltonian cycles? If an additional set of edges is added to P(k,1,…,1) joining C n back to C 0, the resulting graph is still Hamiltonian. Is the same true for other layered graphs? Recognizing layered cubic planar graphs –Are there simple conditions on a graph which guarantee a layered representation? –Is there a fast algorithm for constructing a layering if one exists?