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Minimum leaf number of cubic graphs Gbor Wiener Department of Computer Science and Information Theory Budapest University of Technology and Economics Ghent Graph Theory Workshop, 2017.08.17. Joint work with Jan Goedgebeur, Kenta Ozeki, and

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Minimum leaf number of cubic graphs

Gábor Wiener

Department of Computer Science and Information Theory Budapest University of Technology and Economics Ghent Graph Theory Workshop, 2017.08.17. Joint work with Jan Goedgebeur, Kenta Ozeki, and Nico Van Cleemput

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian. Disproved by Tutte (1946), counterexample of order 46.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian. Disproved by Tutte (1946), counterexample of order 46. Smallest counterexample: order 38.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian. Disproved by Tutte (1946), counterexample of order 46. Smallest counterexample: order 38. Found by Barnette, Bosák, Lederberg (1966). Minimality proved by Holton and McKay (1986).

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian. Disproved by Tutte (1946), counterexample of order 46. Smallest counterexample: order 38. Found by Barnette, Bosák, Lederberg (1966). Minimality proved by Holton and McKay (1986). Conjecture (Barnette, 1969) All 3-connected bipartite planar cubic graphs are hamiltonian.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian. Disproved by Tutte (1946), counterexample of order 46. Smallest counterexample: order 38. Found by Barnette, Bosák, Lederberg (1966). Minimality proved by Holton and McKay (1986).

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian. Disproved by Tutte (1946), counterexample of order 46. Smallest counterexample: order 38. Found by Barnette, Bosák, Lederberg (1966). Minimality proved by Holton and McKay (1986). Conjecture (Barnette, 1969) All 3-connected cubic plane graphs with faces of size at most 6 are hamiltonian.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of planar cubic graphs

Conjecture (Tait, 1880) All 3-connected planar cubic graphs are hamiltonian. Disproved by Tutte (1946), counterexample of order 46. Smallest counterexample: order 38. Found by Barnette, Bosák, Lederberg (1966). Minimality proved by Holton and McKay (1986). Conjecture (Barnette, 1969) All 3-connected cubic plane graphs with faces of size at most 6 are hamiltonian. Proved by F . Kardoš (2014).

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Conjecture (Tutte, 1971) All 3-connected bipartite cubic graphs are hamiltonian.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Conjecture (Tutte, 1971) All 3-connected bipartite cubic graphs are hamiltonian. Disproved by Horton (1976), counterexample of order 96.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Conjecture (Tutte, 1971) All 3-connected bipartite cubic graphs are hamiltonian. Disproved by Horton (1976), counterexample of order 96. Smallest known counterexamples: order 50.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Conjecture (Tutte, 1971) All 3-connected bipartite cubic graphs are hamiltonian. Disproved by Horton (1976), counterexample of order 96. Smallest known counterexamples: order 50. Found by Kelmans (1986). Order of smallest counterexample is between 32 and 50.

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Conjecture (Tutte, 1971) All 3-connected bipartite cubic graphs are hamiltonian. Disproved by Horton (1976), counterexample of order 96. Smallest known counterexamples: order 50. Found by Kelmans (1986). Order of smallest counterexample is between 32 and 50. Connectivity 2:

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Conjecture (Tutte, 1971) All 3-connected bipartite cubic graphs are hamiltonian. Disproved by Horton (1976), counterexample of order 96. Smallest known counterexamples: order 50. Found by Kelmans (1986). Order of smallest counterexample is between 32 and 50. Connectivity 2: several smallest nonhamiltonian examples (planar/non-planar, bipartite/non-bipartite, graph/multigraph) by Asano, Exoo, Harary, Saito (1981) and Asano, Saito (1981).

Gábor Wiener Minimum leaf number of cubic graphs

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Hamiltonicity of cubic graphs

Conjecture (Tutte, 1971) All 3-connected bipartite cubic graphs are hamiltonian. Disproved by Horton (1976), counterexample of order 96. Smallest known counterexamples: order 50. Found by Kelmans (1986). Order of smallest counterexample is between 32 and 50. Connectivity 2: several smallest nonhamiltonian examples (planar/non-planar, bipartite/non-bipartite, graph/multigraph) by Asano, Exoo, Harary, Saito (1981) and Asano, Saito (1981). E.g. the (unique) smallest 2-connected nonhamiltonian cubic planar bipartite graph has order 26.

Gábor Wiener Minimum leaf number of cubic graphs

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Generalizations of traceability

All graphs are undirected, connected, and simple.

Gábor Wiener Minimum leaf number of cubic graphs

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Generalizations of traceability

All graphs are undirected, connected, and simple. Definition The path covering number µ(G) is the minimum number of vertex disjoint paths that cover the vertices of G.

Gábor Wiener Minimum leaf number of cubic graphs

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Generalizations of traceability

All graphs are undirected, connected, and simple. Definition The path covering number µ(G) is the minimum number of vertex disjoint paths that cover the vertices of G. Definition The minimum leaf number ml(G) is the minimum number of leaves (vertices of degree 1) of the spanning trees of G.

Gábor Wiener Minimum leaf number of cubic graphs

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Generalizations of traceability

All graphs are undirected, connected, and simple. Definition The path covering number µ(G) is the minimum number of vertex disjoint paths that cover the vertices of G. Definition The minimum leaf number ml(G) is the minimum number of leaves (vertices of degree 1) of the spanning trees of G. Proposition µ(G) + 1 ≤ ml(G) ≤ 2µ(G).

Gábor Wiener Minimum leaf number of cubic graphs

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Path covering number

Gábor Wiener Minimum leaf number of cubic graphs

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Path covering number

Theorem (Reed, 1996) If G is a cubic graph of order n, then µ(G) ≤ ⌈ n

9⌉.

Gábor Wiener Minimum leaf number of cubic graphs

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Path covering number

Theorem (Reed, 1996) If G is a cubic graph of order n, then µ(G) ≤ ⌈ n

9⌉.

The bound is essentially best possible.

Gábor Wiener Minimum leaf number of cubic graphs

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Path covering number

Theorem (Reed, 1996) If G is a cubic graph of order n, then µ(G) ≤ ⌈ n

9⌉.

The bound is essentially best possible. Conjecture (Reed, 1996) If G is a 2-connected cubic graph of order n, then µ(G) ≤ ⌈ n

10⌉.

Gábor Wiener Minimum leaf number of cubic graphs

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Path covering number

Theorem (Reed, 1996) If G is a cubic graph of order n, then µ(G) ≤ ⌈ n

9⌉.

The bound is essentially best possible. Conjecture (Reed, 1996) If G is a 2-connected cubic graph of order n, then µ(G) ≤ ⌈ n

10⌉.

Confirmed by G. Yu?

Gábor Wiener Minimum leaf number of cubic graphs

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Path covering number

Theorem (Reed, 1996) If G is a cubic graph of order n, then µ(G) ≤ ⌈ n

9⌉.

The bound is essentially best possible. Conjecture (Reed, 1996) If G is a 2-connected cubic graph of order n, then µ(G) ≤ ⌈ n

10⌉.

Confirmed by G. Yu? 2-connected examples with µ(G) =

n 20 (Reed, 1996)

Gábor Wiener Minimum leaf number of cubic graphs

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Path covering number

Theorem (Reed, 1996) If G is a cubic graph of order n, then µ(G) ≤ ⌈ n

9⌉.

The bound is essentially best possible. Conjecture (Reed, 1996) If G is a 2-connected cubic graph of order n, then µ(G) ≤ ⌈ n

10⌉.

Confirmed by G. Yu? 2-connected examples with µ(G) =

n 20 (Reed, 1996)

2-connected examples with µ(G) =

n 14 (G.-O.-V.-W., 2016)

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Theorem (Zoeram-Yaqubi, 2015) If G is a cubic graph of order n, then ml(G) ≤ 2n

9 + 4 9.

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Theorem (Zoeram-Yaqubi, 2015) If G is a cubic graph of order n, then ml(G) ≤ 2n

9 + 4 9.

Conjecture (Zoeram-Yaqubi, 2015) If G is a cubic graph of order n, then ml(G) ≤ n

6 + 1 3.

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Theorem (Zoeram-Yaqubi, 2015) If G is a cubic graph of order n, then ml(G) ≤ 2n

9 + 4 9.

Conjecture (Zoeram-Yaqubi, 2015) If G is a cubic graph of order n, then ml(G) ≤ n

6 + 1 3.

Examples with ml(G) = n

6 + 1 3 (Zoeram-Yaqubi, 2015)

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Theorem (Zoeram-Yaqubi, 2015) If G is a cubic graph of order n, then ml(G) ≤ 2n

9 + 4 9.

Conjecture (Zoeram-Yaqubi, 2015) If G is a cubic graph of order n, then ml(G) ≤ n

6 + 1 3.

Examples with ml(G) = n

6 + 1 3 (Zoeram-Yaqubi, 2015)

Theorem (Salamon-W., 2008) If G is a cubic graph of order n, then ml(G) ≤ n

6 + 4 3.

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Theorem (Boyd-Sitters-van der Ster-Stougie, 2014) If G is a 2-connected cubic multigraph of order n, then ml(G) ≤ n

6 + 2 3.

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Theorem (Boyd-Sitters-van der Ster-Stougie, 2014) If G is a 2-connected cubic multigraph of order n, then ml(G) ≤ n

6 + 2 3.

Proposition If G is a cubic multigraph of order n, then ml(G) ≤ n

4 + 1 2.

Gábor Wiener Minimum leaf number of cubic graphs

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Minimum leaf number

Theorem (Boyd-Sitters-van der Ster-Stougie, 2014) If G is a 2-connected cubic multigraph of order n, then ml(G) ≤ n

6 + 2 3.

Proposition If G is a cubic multigraph of order n, then ml(G) ≤ n

4 + 1 2.

Examples with ml(G) = n

4 + 1 2.

Gábor Wiener Minimum leaf number of cubic graphs

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Main results

Gábor Wiener Minimum leaf number of cubic graphs

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Main results

Theorem (G.-O.-V.-W., 2016) If G is a cubic graph of order n, then ml(G) ≤ n

6 + 1 3.

Gábor Wiener Minimum leaf number of cubic graphs

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Main results

Theorem (G.-O.-V.-W., 2016) If G is a cubic graph of order n, then ml(G) ≤ n

6 + 1 3.

Theorem (G.-O.-V.-W., 2016) If G is a 2-connected cubic graph of order n, then ml(G) ≤ 25n

153 ≈ n 6.12.

Gábor Wiener Minimum leaf number of cubic graphs

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Main results

Theorem (G.-O.-V.-W., 2016) If G is a cubic graph of order n, then ml(G) ≤ n

6 + 1 3.

Theorem (G.-O.-V.-W., 2016) If G is a 2-connected cubic graph of order n, then ml(G) ≤ 25n

153 ≈ n 6.12.

Theorem (G.-O.-V.-W., 2016) If G is a 2-connected cubic graph of order n, then ml(G) ≤ 19n

117 ≈ n 6.157.

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – higher connectivity

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – higher connectivity

Conjecture If G is a 2-connected cubic graph of order n, then ml(G) ≤ ⌈ n

10⌉.

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – higher connectivity

Conjecture If G is a 2-connected cubic graph of order n, then ml(G) ≤ ⌈ n

10⌉.

Conjecture If G is a 3-connected cubic graph of order n, then ml(G) ≤ ⌈ n

16 + 1 2⌉.

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – higher connectivity

Conjecture If G is a 2-connected cubic graph of order n, then ml(G) ≤ ⌈ n

10⌉.

Conjecture If G is a 3-connected cubic graph of order n, then ml(G) ≤ ⌈ n

16 + 1 2⌉.

Conjecture If G is a (2-connected) bipartite cubic graph of order n, then ml(G) ≤ ⌈ n

20⌉ + 1.

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – planar case

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – planar case

No connectivity requirement − → same as the non-planar case.

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – planar case

No connectivity requirement − → same as the non-planar case. Conjecture If G is a 2-connected cubic planar graph of order n, then ml(G) ≤ ⌈ n

14⌉ + 1.

Gábor Wiener Minimum leaf number of cubic graphs

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Conjectures – planar case

No connectivity requirement − → same as the non-planar case. Conjecture If G is a 2-connected cubic planar graph of order n, then ml(G) ≤ ⌈ n

14⌉ + 1.

Question If G is a 3-connected cubic planar graph of order n, then ml(G) ≤ ⌈ n

72 + 1 2⌉?

Gábor Wiener Minimum leaf number of cubic graphs

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Thank you!

Gábor Wiener Minimum leaf number of cubic graphs