Six degrees of graph theory: Kevin Bacon, Paul Erdos,˝ William McKinley and me Ryan Martin [email protected] Assistant Professor Mathematics Department Iowa State University Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 1/61 Joint Work This talk is based on joint work with • Tom Bohman, Carnegie Mellon University • Alan Frieze, Carnegie Mellon University • Michael Krivelevich, Tel Aviv University Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 2/61 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 3/61 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is . Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 3/61 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. This is false. It is Rod Steiger. Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 3/61 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. This is false. It is Rod Steiger. Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 3/61 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. We link two actors together if they appeared together in the same movie. Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 3/61 Six? degrees of separation In the Kevin Bacon Game, it is postulated that the center of the Hollywood universe is Kevin Bacon. We link two actors together if they appeared together in the same movie. (They must be together on a cast list at the IMDb.) Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 3/61 Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to Kevin Bacon. Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 4/61 Bacon number The actor’s # is the fewest number of steps it takes to connect that actor to . Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 4/61 Bacon number The actor’s Bacon number is the fewest number of steps it takes to connect that actor to Kevin Bacon. Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 4/61 Bacon number An actor can have infinite # . Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 4/61 Bacon number An actor can have infinite # . (For example, a TV actor who appears in no movie credits.) Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 4/61 Example: Kevin Costner Kevin Costner is linked to Kevin Bacon because both appeared in JFK (1991). Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 5/61 Example: Kevin Costner is linked to because both appeared in Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 5/61 Example: Kevin Costner is linked to because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is ??? Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 5/61 Example: Kevin Costner is linked to because both appeared in JFK (1991). So, Kevin Costner’s Kevin Bacon number is 1 Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 5/61 Example: Kevin Costner is linked to because both appeared in JFK (1991). # So, ’s is 1 Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 5/61 Illustration: Kevin Costner x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 6/61 Illustration: Kevin Costner x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 6/61 Illustration: Kevin Costner x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 6/61 Illustration: Kevin Costner x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 6/61 E.g.: Henry “Fonz” Winkler We know that • appeared with in Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 7/61 E.g.: Henry “Fonz” Winkler We know that • Henry Winkler appeared with Michael Keaton in Night Shift (1982) • appeared with in Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 7/61 E.g.: Henry “Fonz” Winkler We know that • Henry Winkler appeared with Michael Keaton in Night Shift (1982) • Michael Keaton appeared with Kim Basinger in Batman (1989) • appeared with in Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 7/61 E.g.: Henry “Fonz” Winkler We know that • Henry Winkler appeared with Michael Keaton in Night Shift (1982) • Michael Keaton appeared with Kim Basinger in Batman (1989) • Kim Basinger appeared with Mickey Rourke in 9 1/2 Weeks (1986) • appeared with in Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 7/61 E.g.: Henry “Fonz” Winkler We know that • Henry Winkler appeared with Michael Keaton in Night Shift (1982) • Michael Keaton appeared with Kim Basinger in Batman (1989) • Kim Basinger appeared with Mickey Rourke in 9 1/2 Weeks (1986) • Mickey Rourke appeared with Kevin Bacon in Diner (1982) Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 7/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 A chain: Henry “Fonz” Winkler x x x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 8/61 More: Henry “Fonz” Winkler But it is also true that • appeared with ??? in (2000). Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 9/61 More: Henry “Fonz” Winkler But it is also true that • appeared with in (2000). Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 9/61 More: Henry “Fonz” Winkler But it is also true that • Henry Winkler appeared with Clint Howard in Little Nicky (2000). • appeared with in (2000) Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 9/61 More: Henry “Fonz” Winkler But it is also true that • Henry Winkler appeared with Clint Howard in Little Nicky (2000). • Clint Howard appeared with Kevin Bacon in My Dog Skip (2000). Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 9/61 Better: Henry “Fonz” Winkler x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 10/61 Better: Henry “Fonz” Winkler x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 10/61 Better: Henry “Fonz” Winkler x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 10/61 Better: Henry “Fonz” Winkler x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 10/61 Better: Henry “Fonz” Winkler x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 10/61 Better: Henry “Fonz” Winkler x x x Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 10/61 Can we do even better? has never appeared in a film with . Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p. 11/61 Can we do even better? Kevin Bacon has never appeared in a film with Henry Winkler. # So, ’s is Six degrees of graph theory:Kevin Bacon, Paul Erdos,˝ William McKinley and me – p.