Hypohamiltonian cubic graphs a & Martin Edita M a cajov - PowerPoint PPT Presentation

Hypohamiltonian cubic graphs a & Martin ˇ Edita M´ aˇ cajov´ Skoviera Faculty of Mathematics, Physics and Informatics Comenius University, Bratislava October 8, 2008 Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 1 / 20

Hamiltonian graphs Determining hamiltonicity of cubic graphs is NP complete [Garey, Johnson, Tarjan 1976] Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 2 / 20

Hamiltonian graphs Determining hamiltonicity of cubic graphs is NP complete [Garey, Johnson, Tarjan 1976] Almost all cubic graphs are hamiltonian [Robinson, Wormald 1992] Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 2 / 20

Hamiltonian graphs Determining hamiltonicity of cubic graphs is NP complete [Garey, Johnson, Tarjan 1976] Almost all cubic graphs are hamiltonian [Robinson, Wormald 1992] “nearly” hamiltonian graphs: maximally non-hamiltonian graphs G + e is hamiltonian for every e hypohamiltonian (HH) G − v is hamiltonian for every v Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 2 / 20

✎✏ ✌✍ �✁ ✂✄ ☎✆ ✝✞ ✟✠ Hypohamiltonian graphs Petersen graph is the smallest among HH ✓☛✓ ✔☛✔ ☞☛☞ ✡☛✡ ✑☛✑✒ Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 3 / 20

�✁ ✂✄ ☎✆ ✝✞ ✟✠ ✌✍ ✎✏ Hypohamiltonian graphs Petersen graph is the smallest among HH ✔☛✔ ✓☛✓ ✡☛✡ ☞☛☞ ✑☛✑✒ each HH graph is 3-edge-connected ⇒ minimum valency is 3 Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 3 / 20

✎✏ �✁ ✂✄ ☎✆ ✝✞ ✟✠ ✌✍ Hypohamiltonian graphs Petersen graph is the smallest among HH ✔☛✔ ✓☛✓ ☞☛☞ ✡☛✡ ✑☛✑✒ each HH graph is 3-edge-connected ⇒ minimum valency is 3 no HH graph can be bipartite Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 3 / 20

Hypohamiltonian graphs explicit constructions by Sousellier, Lindgren (1967), Bondy (1972), Chv´ atal (1973), . . . Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 4 / 20

Hypohamiltonian graphs explicit constructions by Sousellier, Lindgren (1967), Bondy (1972), Chv´ atal (1973), . . . ∃ a HH graph of order n for each n ≥ 18 [Chv´ atal 1973, Aldred, McKay, Wormald 1997] Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 4 / 20

Hypohamiltonian graphs explicit constructions by Sousellier, Lindgren (1967), Bondy (1972), Chv´ atal (1973), . . . ∃ a HH graph of order n for each n ≥ 18 [Chv´ atal 1973, Aldred, McKay, Wormald 1997] exponentially many HH graphs of order n [Collier, Schmeichel 1972] Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 4 / 20

Hypohamiltonian graphs explicit constructions by Sousellier, Lindgren (1967), Bondy (1972), Chv´ atal (1973), . . . ∃ a HH graph of order n for each n ≥ 18 [Chv´ atal 1973, Aldred, McKay, Wormald 1997] exponentially many HH graphs of order n [Collier, Schmeichel 1972] Thomassen’s construction (1974) Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 4 / 20

Hypohamiltonian graphs explicit constructions by Sousellier, Lindgren (1967), Bondy (1972), Chv´ atal (1973), . . . ∃ a HH graph of order n for each n ≥ 18 [Chv´ atal 1973, Aldred, McKay, Wormald 1997] exponentially many HH graphs of order n [Collier, Schmeichel 1972] Thomassen’s construction (1974) Chv´ atal (1972): Does there ∃ a planar hypohamiltonian graph? Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 4 / 20

Hypohamiltonian graphs explicit constructions by Sousellier, Lindgren (1967), Bondy (1972), Chv´ atal (1973), . . . ∃ a HH graph of order n for each n ≥ 18 [Chv´ atal 1973, Aldred, McKay, Wormald 1997] exponentially many HH graphs of order n [Collier, Schmeichel 1972] Thomassen’s construction (1974) Chv´ atal (1972): Does there ∃ a planar hypohamiltonian graph? infinitely many planar hypohamiltonian graphs (order ≥ 105) [Thomassen 1976] Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 4 / 20

✾✆✾✿ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✗✆✗ ✘✆✘ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✘✆✘ ✗✆✗ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ❀✆❀❁ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✗✆✗ ✘✆✘ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ❂✆❂❃ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✸✆✸✆✸✆✸✆✸✆✸✆✸ ✵✆✵✆✵✆✵✆✵✆✵✆✵ ✴✆✴✆✴✆✴✆✴✆✴✆✴ ✹✆✹✆✹✆✹✆✹✆✹✆✹ ✲✆✲ ✤✆✤✆✤ ✥✆✥✆✥ ✒✆✒ ✖✆✖✆✖ ✕✆✕✆✕ ✳✆✳ ❊✆❊❋ ❨✆❨❩ ✍✆✍✆✍ ✞✆✞✆✞✆✞ ✟✆✟✆✟ ✠✆✠✆✠✆✠ ✡✆✡✆✡ ✻✆✻✆✻ ✺✆✺✆✺ ✑✆✑ ✷✆✷✆✷ ✶✆✶✆✶ ✞✆✞✆✞✆✞ ✟✆✟✆✟ ✠✆✠✆✠✆✠ ✡✆✡✆✡ ✺✆✺✆✺ ✻✆✻✆✻ ✷✆✷✆✷ ✶✆✶✆✶ ✞✆✞✆✞✆✞ ✟✆✟✆✟ ✠✆✠✆✠✆✠ ✡✆✡✆✡ ✻✆✻✆✻ ✺✆✺✆✺ ✶✆✶✆✶ ✷✆✷✆✷ ✞✆✞✆✞✆✞ ✠✆✠✆✠✆✠ ✶✆✶✆✶ ✟✆✟✆✟ ✡✆✡✆✡ ✻✆✻✆✻ ✺✆✺✆✺ ✷✆✷✆✷ ✞✆✞✆✞✆✞ ✟✆✟✆✟ ✠✆✠✆✠✆✠ ✡✆✡✆✡ ✺✆✺✆✺ ✻✆✻✆✻ ✶✆✶✆✶ ✷✆✷✆✷ ✞✆✞✆✞✆✞ ✟✆✟✆✟ ✠✆✠✆✠✆✠ ✡✆✡✆✡ ✺✆✺✆✺ ✻✆✻✆✻ ✷✆✷✆✷ ✶✆✶✆✶ ✞✆✞✆✞✆✞ ✟✆✟✆✟ ✠✆✠✆✠✆✠ ✡✆✡✆✡ ✺✆✺✆✺ ✻✆✻✆✻ ✷✆✷✆✷ ✶✆✶✆✶ ✻✆✻✆✻ ✺✆✺✆✺ ✶✆✶✆✶ ✷✆✷✆✷ ✶✆✶✆✶ ✺✆✺✆✺ ✻✆✻✆✻ ✷✆✷✆✷ ✻✆✻✆✻ ✺✆✺✆✺ ✶✆✶✆✶ ✷✆✷✆✷ ✻✆✻✆✻ ✺✆✺✆✺ ✷✆✷✆✷ ✶✆✶✆✶ ✻✆✻✆✻ ✺✆✺✆✺ ✪✆✪ ✶✆✶✆✶ ✷✆✷✆✷ ●✆●❍ ✱✆✱ ✰✆✰ ✺✆✺✆✺ ✻✆✻✆✻ ✪✆✪ ✷✆✷✆✷ ✶✆✶✆✶ ✰✆✰ ✱✆✱ ✪✆✪ ☎✆☎✝ ✺✆✺✆✺ ✻✆✻✆✻ ✦✆✦ ✧✆✧ ✷✆✷✆✷ ✶✆✶✆✶ ✌✆✌✆✌✆✌✆✌✆✌✆✌✆✌✆✌✆✌ ✺✆✺✆✺ ■✆■❏ ✱✆✱ ✬✆✬ ✰✆✰ ✪✆✪ ✶✆✶✆✶ ✷✆✷✆✷ ✻✆✻✆✻ ✯✆✯ ✮✆✮ ✭✆✭ ★✆★ ✧✆✧ ✦✆✦ ❑✆❑▲ ❙✆❙❚ ✬✆✬ ✭✆✭ ✶✆✶✆✶ ✷✆✷✆✷ ★✆★ ✬✆✬ ✭✆✭ ★✆★ ▼✆▼◆ ☞ ☞ ☛ ☞ ☞ ☞ ❯❱ ☞ ☞ ☞ ☞ ☛ ☛ ☛ ☛ ☛ ☛ ✼✽ ☛ ☛ ❄❅ ❆❇ ❈❉ ✂✄ ✓ �✁ ❪❫ ❖P ◗❘ ✎✏ ✙ ✓ ✩ ✢ ✢ ✣ ✣ ✣ ✣ ✩ ✢ ✩ ✫ ✫ ✫ ✫ ❬❭ ✢ ✜ ✓ ✙ ✓ ✔ ✔ ✔ ✔ ✙ ✚ ✜ ✚ ✚ ✛ ✛ ✛ ✜ ❲❳ Cubic hypohamiltonian graphs Petersen (10), Lindgren (16), Blanuˇ sa snarks (18), . . . Edita M´ aˇ cajov´ a (Bratislava) HH cubic graphs October 8, 2008 5 / 20