# Connections between decomposition trees of 3-connected plane triangulations and hamiltonian properties

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1 Introduction Hamiltonicity Other hamiltonian properties Connections between decomposition trees of 3-connected plane triangulations and hamiltonian properties Gunnar Brinkmann Jasper Souffriau Nico Van Cleemput Ghent University

2 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Triangulation A triangulation is a plane graph in which each face is a triangle.

3 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Hamiltonian cycle A hamiltonian cycle in G(V, E) is a subgraph of G(V, E) which is isomorphic to C V. A graph is hamiltonian if it contains a hamiltonian cycle.

4 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Separating triangles A separating triangle S in a triangulation T is a subgraph of T such that S is isomorphic to C 3 and T S has two components.

5 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness 4-connected triangulations A triangulation is 4-connected if and only if it contains no separating triangles.

6 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Whitney Theorem (Whitney, 1931) Each triangulation without separating triangles is hamiltonian.

7 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Splitting triangulations

8 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Recursively splitting triangulations 4-connected parts

9 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Decomposition tree Vertices: 4-connected parts Edges: separating triangles

10 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Decomposition trees and hamiltonicity For each tree T there exist hamiltonian triangulations which have T as decomposition tree. A triangulation G with decomposition tree T is hamiltonian if... Whitney (1930): E(T ) = 0 Thomassen (1978), Chen (2003): E(T ) 1 Böhme, Harant, Tkáč (1993): E(T ) 2 Jackson, Yu (2002): (T ) 3

11 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Jackson and Yu (T ) 4 is not sufficient to imply hamiltonicity. hamiltonian hamiltonian not hamiltonian

12 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Question Can the result of Jackson and Yu be improved? Which trees can arise as decomposition trees of non-hamiltonian triangulations?

13 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Subdividing a face with a graph

14 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Subdividing a face with a graph

15 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Subdividing a non-hamiltonian triangulation Lemma When a non-hamiltonian triangulation is subdivided, then the resulting graph is also non-hamiltonian.

16 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Toughness A graph is 1-tough if it cannot be split into k components by removing less than k vertices.

17 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Toughness A hamiltonian graph is 1-tough.

18 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Graphs that are not 1-tough are trivially non-hamiltonian. Remove 4 black and 4 red vertices 12 blue components remain

19 Introduction Hamiltonicity Other hamiltonian properties Definitions Decomposition Constructions Toughness Creating a non-hamiltonian plane graph Lemma When in a plane graph with more faces than vertices each face is subdivided, then the resulting plane graph is non-hamiltonian. The subdivided graph is not 1-tough.

20 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Decomposition trees with 6 Theorem For each tree T with (T ) 6, there exists a non-hamiltonian triangulation G, such that T is the decomposition tree of G. Constructive proof.

21 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Assume (T ) = 6. T 1 T 2 T 3 T 4 T 5 T 6 Choose triangulation G i with decomposition tree T i (1 i 6)

22 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 G 4 G 5 G 6 G 1 G 3 G 2 A non-hamiltonian triangulation with T as decomposition tree.

23 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 (T ) > 6

24 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Remaining cases :

25 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Remaining cases : Not the decomposition tree of non-hamiltonian triangulation

26 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Remaining cases : Not the decomposition tree of non-hamiltonian triangulation Possibly the decomposition tree of non-hamiltonian triangulation

27 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Remaining cases : Not the decomposition tree of non-hamiltonian triangulation? Possibly the decomposition tree of non-hamiltonian triangulation

28 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Multiple degrees > 3 Theorem For each tree T with at least two vertices with degree > 3, there exists a non-hamiltonian triangulation G, such that T is the decomposition tree of G.

29 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 T 4 T 1 T k T 8 T 1 T5 T 2 T 6 T 3 k subcubic vertices T 7... G 1 G k G 4 G 8 G 1 G 2 G 3 G 5 G 6 G 7 red vertices: 5 + k + (5 3) = 7 + k components: 4 + k + 4 = 8 + k

30 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Remaining cases: trees with one vertex of degree 4 or 5 and all other degrees at most 3.

31 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 One vertex of degree 4 or 5 Theorem Let G be a triangulation with decomposition tree T with only one vertex of degree 4 or 5 and all other vertices of degree at most 3. Then G is 1-tough.

32 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Theorem (Jackson and Yu, 2002) Let G be a triangulation with decomposition tree T, (T ) 3 and uvw a facial triangle of G that is also a facial triangle in a vertex of T with degree at most 2. Then G has a hamiltonian cycle through uv and vw. u w v

33 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 This implies: Non-hamiltonian triangulations with decomposition trees with one vertex of degree k 4 and all others of degree at most 3 exists if and only if... non-hamiltonian triangulations with decomposition tree K 1,k exist. 4-connected triangulations exist with facial triangles t 1,..., t k so that no hamiltonian cycle C and distinct edges e 1,..., e k C exist such that e i t i. Also valid for k {1, 2, 3}

34 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5 Specialised search Lemma All triangulations on at most 31 vertices with K 1,4 as decomposition tree are hamiltonian. Lemma All triangulations on at most 27 vertices with K 1,5 as decomposition tree are hamiltonian.

35 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5... and now? Prove that for each 4-tuple of vertex-disjoint triangles in a 4-connected triangulation there exists a hamiltonian cycle that shares an edge with each of the triangles. or Find a counterexample.

36 Introduction Hamiltonicity Other hamiltonian properties 6 Multiple degrees > 3 One vertex of degree 4 or 5... and now? Prove that for each 5-tuple of triangles T 1, T 2, T 3, T 4, T 5 in a 4-connected triangulation there exists a hamiltonian cycle C and distinct edges e 1, e 2, e 3, e 4, e 5 C such that e i T i. or Find a counterexample.

37 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Hamiltonian path A hamiltonian path in G(V, E) is a subgraph of G(V, E) which is isomorphic to P V.

38 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Traceable A graph G(V, E) is traceable if it contains a hamiltonian path.

39 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Decomposition trees with 8 Theorem For each tree T with (T ) 8, there exists a non-traceable triangulation G, such that T is the decomposition tree of G. s 8 vertices

40 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Decomposition trees with {6, 7} Theorem For each tree T with a pair of vertices with degrees k 1 and k 2 with (k 1, k 2 ) {(6, 4), (6, 5), (6, 6), (7, 4), (7, 5), (7, 6), (7, 7)} and all others of degree at most 3, there exists a non-traceable triangulation G, such that T is the decomposition tree of G.

41 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Decomposition trees with = 4 Theorem Let T be a tree with one vertex of degree 4 and all others of degree at most 3. Then any triangulation which has T as decomposition tree is traceable.

42 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Decomposition trees with = 4

43 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Hamiltonian-connected A hamiltonian path connecting x and y is a hamiltonian path P such that x and y have degree 1 in P. A graph G(V, E) is hamiltonian-connected if for each pair x, y of distinct vertices in V there exists a hamiltonian path connecting x and y.

44 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview 4-connected triangulations Theorem (Thomassen, 1983) Each triangulation without separating triangles is hamiltonian-connected.

45 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview 3-connected triangulations Theorem Let G be a 3-connected triangulation such that there is an edge e which is contained in all separating triangles. Then G is hamiltonian-connected.

46 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Decomposition tree Theorem Let T be a tree with maximum degree 1. Then any triangulation which has T as decomposition tree is hamiltonian-connected. Theorem Let T be a tree with maximum degree at least 4. Then T is the decomposition tree of a 3-connected triangulation which is not hamiltonian-connected. 4 vertices

47 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Decomposition tree Lemma On up to 21 vertices all triangulations that have a decomposition tree with maximum degree 2 are all hamiltonian-connected. Lemma On up to 20 vertices all triangulations that have a decomposition tree with maximum degree 3 are all hamiltonian-connected.

48 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Summary for hamiltonian-connectedness :

49 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Summary for hamiltonian-connectedness : Always hamiltonianconnected

50 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Summary for hamiltonian-connectedness : Always hamiltonianconnected Possibly not hamiltonian-connected

51 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Summary for hamiltonian-connectedness : Always hamiltonianconnected? Possibly not hamiltonian-connected

52 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Overview Definition A graph G is k-edge-hamiltonian-connected if for any X {x 1 x 2 : x 1, x 2 V (G), x 1 x 2 } such that 1 X k and X is a forest of paths, G X has a hamiltonian cycle containing all edges in X. 1-edge-hamiltonian-connected is equivalent to hamiltonianconnected. Definition A graph G is k-hamiltonian if for any k vertices v 1,..., v k in G, G {v 1,..., v k } is hamiltonian.

53 Introduction Hamiltonicity Other hamiltonian properties Traceable Hamiltonian-connected Overview Table 0 1,1 2,... 3,... 4,( 3),... 4,4,... 5,( 3),... 5,4,... 5,5,... 6,( 3),... 6,4,... 6,5,... 6,6,... 7,( 3),... 7,4,... 7,5,... 7,6,... 7,7,... 8,... Traceable Hamiltonian Hamiltonian- 2-edge- 1-hamiltonian 2-hamiltonian connected hamiltonian

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