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, Generation of hypohamiltonian graphs Jan Goedgebeur* Carol T. Zamfirescu * Combinatorial Algorithms and Algorithmic Graph Theory Department of Applied Mathematics and Computer Science Ghent University, Belgium caagt Jan Goedgebeur, Carol

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Generation of hypohamiltonian graphs

Jan Goedgebeur* Carol T. Zamfirescu

* Combinatorial Algorithms and Algorithmic Graph Theory Department of Applied Mathematics and Computer Science Ghent University, Belgium

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 1

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Construction of hypohamiltonian graphs

Definition A graph G is hypohamiltonian if G is non-hamiltonian and G − v is hamiltonian for every v ∈ V(G).

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 2

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Construction of hypohamiltonian graphs

Definition A graph G is hypohamiltonian if G is non-hamiltonian and G − v is hamiltonian for every v ∈ V(G). Introduced by Sousselier in 1963. Smallest hypohamiltonian graph: Petersen graph.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 2

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Construction of hypohamiltonian graphs

Chvátal, 1973:

There exists a hypohamiltonian graph for every order n ≥ 26.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 3

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Construction of hypohamiltonian graphs

Chvátal, 1973:

There exists a hypohamiltonian graph for every order n ≥ 26.

Aldred, McKay, and Wormald, 1997:

Generation algorithm for hypohamiltonian graphs. Determined all hypohamiltonian graphs up to 17 vertices. Order 10 11 12 13 14 15 16 17 # graphs 1 1 1 4

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 3

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Construction of hypohamiltonian graphs

Chvátal, 1973:

There exists a hypohamiltonian graph for every order n ≥ 26.

Aldred, McKay, and Wormald, 1997:

Generation algorithm for hypohamiltonian graphs. Determined all hypohamiltonian graphs up to 17 vertices. Order 10 11 12 13 14 15 16 17 # graphs 1 1 1 4

Combined with other results:

There exists a hypohamiltonian graph of order n ⇐ ⇒ n ∈ {10, 13, 15, 16} or n ≥ 18.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 3

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Construction of planar hypohamiltonian graphs

In 1972 Chvátal asked whether planar hypohamiltonian graphs exist.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 4

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Construction of planar hypohamiltonian graphs

In 1972 Chvátal asked whether planar hypohamiltonian graphs exist. Grünbaum conjectured their nonexistence.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 4

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Construction of planar hypohamiltonian graphs

In 1972 Chvátal asked whether planar hypohamiltonian graphs exist. Grünbaum conjectured their nonexistence. Thomassen, 1976:

There are infinitely many planar hypohamiltonian graphs. (Thomassen’s smallest example has 105 vertices)

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 4

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Smallest planar hypohamiltonian graph

Search for the smallest planar hypohamiltonian graph:

1976: Thomassen: 105 vertices.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 5

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Smallest planar hypohamiltonian graph

Search for the smallest planar hypohamiltonian graph:

1976: Thomassen: 105 vertices. 1979: Hatzel: 57 vertices. 2007: T. Zamfirescu and C.T. Zamfirescu: 48 vertices. 2011: Araya and Wiener: 42 vertices.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 5

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Smallest planar hypohamiltonian graph

Search for the smallest planar hypohamiltonian graph:

1976: Thomassen: 105 vertices. 1979: Hatzel: 57 vertices. 2007: T. Zamfirescu and C.T. Zamfirescu: 48 vertices. 2011: Araya and Wiener: 42 vertices. (All by hand)

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 5

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Smallest planar hypohamiltonian graph

Theorem (Jooyandeh, McKay, Östergård, Pettersson and Zamfirescu,

2016)

There exist at least 25 planar hypohamiltonian graphs on 40 vertices.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 6

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Smallest planar hypohamiltonian graph

Theorem (Jooyandeh, McKay, Östergård, Pettersson and Zamfirescu,

2016)

There exist at least 25 planar hypohamiltonian graphs on 40 vertices.

Lower bound smallest planar hypohamiltonian graph?

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 6

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Smallest planar hypohamiltonian graph

Theorem (Jooyandeh, McKay, Östergård, Pettersson and Zamfirescu,

2016)

There exist at least 25 planar hypohamiltonian graphs on 40 vertices.

Lower bound smallest planar hypohamiltonian graph?

Theorem (Aldred, McKay, and Wormald, 1997) The smallest planar hypohamiltonian graph has at least 18 vertices.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 6

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Generation of hypohamiltonian graphs Our aim:

Faster generation algorithm for hypohamiltonian graphs. Based on algorithm of Aldred, McKay, and Wormald (1997). Including support for:

Generation of hypohamiltonian graphs with a lower bound

  • n the girth.

Generation of planar hypohamiltonian graphs.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 7

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Generation of hypohamiltonian graphs

Basic algorithm to generate all hypohamiltonian graphs with n vertices (simplified):

Start from Cn−1 + v Recursively add edges in all possible ways

Output hypohamiltonian graphs.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 8

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Properties of hypohamiltonian graphs

Lemma (Bondy, 1972) Let G be a hypohamiltonian graph containing a triangle T. Then every vertex of T has degree at least 4.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 9

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Properties of hypohamiltonian graphs

Lemma (Bondy, 1972) Let G be a hypohamiltonian graph containing a triangle T. Then every vertex of T has degree at least 4.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 9

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Properties of hypohamiltonian graphs

Lemma (Bondy, 1972) Let G be a hypohamiltonian graph containing a triangle T. Then every vertex of T has degree at least 4.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 9

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Properties of hypohamiltonian graphs

Lemma (Bondy, 1972) Let G be a hypohamiltonian graph containing a triangle T. Then every vertex of T has degree at least 4.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 9

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Properties of hypohamiltonian graphs

Lemma (Bondy, 1972) Let G be a hypohamiltonian graph containing a triangle T. Then every vertex of T has degree at least 4. Other forbidden configurations...

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 9

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Properties of hypohamiltonian graphs

Lemma (Aldred, McKay, and Wormald, 1997) Given a hypocyclic graph G, for any partition (W, X) of the vertices of G with |W| > 1 and |X| > 1, we have that p(G[W]) < |X| and k(G[W]) < |X| Lemma (Aldred, McKay, and Wormald, 1997) Let G be a hypocyclic graph, and consider a partition (W, X)

  • f the vertices of G with |W| > 1 and |X| > 1 such that W is

an independent set. Then we have 2n2 + n1 ≥ 2|W| for every v ∈ X. This are called type A, B and C obstructions.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 10

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Properties of hypohamiltonian graphs

Lemma (Aldred, McKay, and Wormald, 1997) Given a hypocyclic graph G, for any partition (W, X) of the vertices of G with |W| > 1 and |X| > 1, we have that p(G[W]) < |X| and k(G[W]) < |X| Lemma (Aldred, McKay, and Wormald, 1997) Let G be a hypocyclic graph, and consider a partition (W, X)

  • f the vertices of G with |W| > 1 and |X| > 1 such that W is

an independent set. Then we have 2n2 + n1 ≥ 2|W| for every v ∈ X. This are called type A, B and C obstructions. We refined these lemmas and added additional lemmas.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 10

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Generation of hypohamiltonian graphs

Basic algorithm to generate all hypohamiltonian graphs with n vertices:

Start from Cn−1 + v Recursively add edges in all possible ways

Output hypohamiltonian graphs.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 11

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Specialised construction algorithm

if Graph is non-hamiltonian AND not generated before then if Graph contains triangle with vertex v of degree < 4 then Add a valid edge which increases the degree of v in all possible ways else if Graph contains ... (i.e. try to apply other lemmas) then Destroy ... in all possible ways else if Graph is hypohamiltonian then Output graph end if Add one valid edge in all possible ways end if end if

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 12

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Specialised construction algorithm

if Graph is non-hamiltonian AND not generated before then if Graph contains triangle with vertex v of degree < 4 then Add a valid edge which increases the degree of v in all possible ways else if Graph contains ... (i.e. try to apply other lemmas) then Destroy ... in all possible ways else if Graph is hypohamiltonian then Output graph end if Add one valid edge in all possible ways end if end if

Note: our algorithm never removes any edges. Useful for planarity / girth !

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 12

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Results – hypohamiltonian graphs

Order # hypoham. g = 3 g = 4 g = 5 g = 6 g = 7 g ≥ 8 0 − 9 10 1 1 11 12 13 1 1 14 15 1 1 16 4 4 17 18 14 1 5 8 19 34 34 20 ? ? ? 4 21 ? ? ? 85 22 ? ? ? 420 23 ? ? ? ? 24 ? ? ? ? 25 ? ? ? ? 1 26 ? ? ? ? 27 ? ? ? ? ? 28 ? ? ? ? ≥ 1 1 29 ? ? ? ? ? 30 ? ? ? ? ? 31 − 35 ? ? ? ? ? ?

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 13

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Results – running times

Order # hypoham. Time (s) Increase 16 4 9 17 189 21.00 18 14 18 339 97.03 19 34 ≈ 400

Table: Counts and generation times for hypohamiltonian graphs.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 14

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Results – hypohamiltonian graphs on 18 vertices

[Can be downloaded from the House of Graphs (http://hog.grinvin.org)]

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 15

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Results – hypohamiltonian graphs

Theorem (JG and Zamfirescu) The smallest hypohamiltonian graph of girth 6 (girth 7) has

  • rder 25 (order 28).

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 16

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Results – planar hypohamiltonian graphs

Theorem (JG and Zamfirescu) The smallest planar hypohamiltonian graph has at least 23 vertices, and the smallest planar hypohamiltonian graph of girth 4 has at least 25 vertices.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 17

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Results – planar hypohamiltonian graphs

Theorem (JG and Zamfirescu) The smallest planar hypohamiltonian graph has at least 23 vertices, and the smallest planar hypohamiltonian graph of girth 4 has at least 25 vertices.

Combined with upper bounds:

Girth 3: 23 ≤ n ≤ 216 (Zamfirescu) Girth 4: 25 ≤ n ≤ 40 (Jooyandeh et al.)

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 17

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Results – planar hypohamiltonian graphs

Theorem (JG and Zamfirescu) The smallest planar hypohamiltonian graph has at least 23 vertices, and the smallest planar hypohamiltonian graph of girth 4 has at least 25 vertices.

Combined with upper bounds:

Girth 3: 23 ≤ n ≤ 216 (Zamfirescu) Girth 4: 25 ≤ n ≤ 40 (Jooyandeh et al.) Girth 5: n = 45 (Jooyandeh et al.)

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 17

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Results – cubic planar hypohamiltonian graphs

Question (Chvátal, 1973) Do cubic planar hypohamiltonian graphs exist?

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 18

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Results – cubic planar hypohamiltonian graphs

Question (Chvátal, 1973) Do cubic planar hypohamiltonian graphs exist? Theorem (Thomassen, 1981) Cubic planar hypohamiltonian graphs exist. (Infinite family with 94 + 4k vertices.)

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 18

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Results – cubic planar hypohamiltonian graphs

Question (Chvátal, 1973) Do cubic planar hypohamiltonian graphs exist? Theorem (Thomassen, 1981) Cubic planar hypohamiltonian graphs exist. (Infinite family with 94 + 4k vertices.) Theorem (Zamfirescu, 2015) There exist cubic planar hypohamiltonian graphs of order n for every even n ≥ 74.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 18

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Results – cubic planar hypohamiltonian graphs

Question (Chvátal, 1973) Do cubic planar hypohamiltonian graphs exist? Theorem (Thomassen, 1981) Cubic planar hypohamiltonian graphs exist. (Infinite family with 94 + 4k vertices.) Theorem (Zamfirescu, 2015) There exist cubic planar hypohamiltonian graphs of order n for every even n ≥ 74. The smallest known cubic planar hypohamiltonian graph has 70 vertices (Araya and Wiener, 2011).

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 18

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Results – cubic planar hypohamiltonian graphs

Question (Chvátal, 1973) Do cubic planar hypohamiltonian graphs exist? Theorem (Thomassen, 1981) Cubic planar hypohamiltonian graphs exist. (Infinite family with 94 + 4k vertices.) Theorem (Zamfirescu, 2015) There exist cubic planar hypohamiltonian graphs of order n for every even n ≥ 74. The smallest known cubic planar hypohamiltonian graph has 70 vertices (Araya and Wiener, 2011). All of these known graphs have girth 4...

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 18

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Results – cubic planar hypohamiltonian graphs

Theorem (McKay, 2016) The smallest cubic planar hypohamiltonian graphs of girth 5 have order 76.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 19

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Results – cubic planar hypohamiltonian graphs

Theorem (McKay, 2016) The smallest cubic planar hypohamiltonian graphs of girth 5 have order 76. (Exactly three such graphs, all with a trivial automorphism group.)

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 19

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Results – cubic planar hypohamiltonian graphs

Theorem (McKay, 2016) The smallest cubic planar hypohamiltonian graphs of girth 5 have order 76. (Exactly three such graphs, all with a trivial automorphism group.) Question (McKay, 2016) Are there infinitely many cubic planar hypohamiltonian graphs

  • f girth 5?

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 19

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Results – cubic planar hypohamiltonian graphs

Theorem (JG and Zamfirescu) Let G be a cubic non-hamiltonian graph and let H be a cubic hypohamiltonian graph. If G contains a pair of good edges, then the dot product G · H applied to these good edges is a hypohamiltonian graph.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 20

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Results – cubic planar hypohamiltonian graphs

By applying this “good edges” theorem:

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 21

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Results – cubic planar hypohamiltonian graphs

By applying this “good edges” theorem: Theorem (JG and Zamfirescu) There are infinitely many planar cubic hypohamiltonian graphs

  • f girth 5.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 21

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Results – cubic planar hypohamiltonian graphs

By applying this “good edges” theorem: Theorem (JG and Zamfirescu) There are infinitely many planar cubic hypohamiltonian graphs

  • f girth 5.

Theorem (JG and Zamfirescu) There exists a hypohamiltonian snark of order n if and only if n ∈ {10, 18, 20, 22} or n is even and n ≥ 26.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 21

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Results – cubic planar hypohamiltonian graphs

By using the program plantri were able to show that:

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 22

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Results – cubic planar hypohamiltonian graphs

By using the program plantri were able to show that: The smallest cubic planar hypohamiltonian graph has at least 54 vertices. (Recall: upper bound 70 vertices).

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 22

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Results – cubic planar hypohamiltonian graphs

By using the program plantri were able to show that: The smallest cubic planar hypohamiltonian graph has at least 54 vertices. (Recall: upper bound 70 vertices). There is exactly one cubic planar hypohamiltonian graph

  • f order 78 and girth 5. It has D3h symmetry.

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 22

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Thanks for your attention!

Jan Goedgebeur, Carol T. Zamfirescu Generation of hypohamiltonian graphs 23