Pairs of Forbidden Subgraphs for Pancyclicity

Transcription

1 1 / 24 Pairs of Forbidden Subgraphs for Pancyclicity James Carraher University of Nebraska Lincoln s-jcarrah1@math.unl.edu Joint Work with Mike Ferrara, Tim Morris, Mike Santana October 2012

2 Hamiltonian 2 / 24 Definition: A hamiltonian cycle is a cycle that contains every vertex. A graph G is hamiltonian if it has a hamiltonian cycle.

3 Hamiltonian 2 / 24 Definition: A hamiltonian cycle is a cycle that contains every vertex. A graph G is hamiltonian if it has a hamiltonian cycle.

4 Hamiltonian 3 / 24 Determining if a graph is hamiltonian is NP-complete. Some sufficient conditions that imply hamiltonian. Dirac s Theorem: Let G be an n-vertex graph. If n 3 and δ(g) n/2 then G is hamiltonian. Ore s Theorem: Let G be an n-vertex graph. If n 3 and d() + d() n whenever / E(G), then G is hamiltonian. We are interested in conditions of connectivity and avoiding certain subgraphs implying the graph is hamiltonian.

5 4 / 24 Hamiltonian Forbidden Subgraphs Definition: A graph G is said to be H-free if it does not contain a copy of H as an induced subgraph. claw paw

6 4 / 24 Hamiltonian Forbidden Subgraphs Definition: A graph G is said to be H-free if it does not contain a copy of H as an induced subgraph. claw paw

7 5 / 24 Hamiltonian Theorem (Goodman Hedetniemi, 1974) If G is a 2-connected graph that is claw-free and paw-free, then G is hamiltonian. Claw Paw

8 5 / 24 Hamiltonian Theorem (Goodman Hedetniemi, 1974) If G is a 2-connected graph that is claw-free and paw-free, then G is hamiltonian. Claw Paw

9 5 / 24 Hamiltonian Theorem (Goodman Hedetniemi, 1974) If G is a 2-connected graph that is claw-free and paw-free, then G is hamiltonian. Claw Paw

10 5 / 24 Hamiltonian Theorem (Goodman Hedetniemi, 1974) If G is a 2-connected graph that is claw-free and paw-free, then G is hamiltonian. Claw Paw

11 5 / 24 Hamiltonian Theorem (Goodman Hedetniemi, 1974) If G is a 2-connected graph that is claw-free and paw-free, then G is hamiltonian. Claw Paw

12 5 / 24 Hamiltonian Theorem (Goodman Hedetniemi, 1974) If G is a 2-connected graph that is claw-free and paw-free, then G is hamiltonian. Claw Paw

13 6 / 24 Hamiltonian Matthews, Sumner Conjecture Matthews, Sumner Conjecture: (1984) Every 4-connected claw-free graph is hamiltonian. There has been a great deal of work done to prove this conjecture. One of many notable results is the Ryjá cek closure. Theorem: (Kaiser, Vrána, 2010) Every 5-connected claw-free graph with minimum degree at least 6 is hamiltonian.

14 7 / 24 Generalized Net Hamiltonian We are interested in graphs that are claw-free that also avoid another subgraph. Definition: The generalized net N(, j, k) is a triangle with three pendant paths of length, j and k. N(2, 2, 2)

15 Hamiltonian 8 / 24 Theorem: (Faudree, Gould, 1997, extending Bedrossian 1991) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 2-connected graph of order n 10. G is {X, Y}-free implies G is hamiltonian if and only if X is the claw and Y is an induced subgraph of one of the following. N(1, 1, 1) N(2, 1, 0) N(3, 0, 0) P 6

16 Hamiltonian 9 / 24 Forbidden Subgraphs for higher connectivity. Theorem: (Łuczak, Pfender, 2004) Every 3-connected claw-free P 11 -free graph is hamiltonian. Theorem: (Lai, Xiong, Yan, Yan, 2009) Every 3-connected claw-free N(8, 0, 0)-free graph is hamiltonian.

17 10 / 24 Pancyclicity We want to prove a similar theorem to the previous result for pancyclic, a stronger condition then hamiltonian. Definition: A graph G on n vertices is said to be pancyclic if it contains a cycle of length from 3 to n.

18 10 / 24 Pancyclicity We want to prove a similar theorem to the previous result for pancyclic, a stronger condition then hamiltonian. Definition: A graph G on n vertices is said to be pancyclic if it contains a cycle of length from 3 to n.

19 10 / 24 Pancyclicity We want to prove a similar theorem to the previous result for pancyclic, a stronger condition then hamiltonian. Definition: A graph G on n vertices is said to be pancyclic if it contains a cycle of length from 3 to n.

20 10 / 24 Pancyclicity We want to prove a similar theorem to the previous result for pancyclic, a stronger condition then hamiltonian. Definition: A graph G on n vertices is said to be pancyclic if it contains a cycle of length from 3 to n.

21 10 / 24 Pancyclicity We want to prove a similar theorem to the previous result for pancyclic, a stronger condition then hamiltonian. Definition: A graph G on n vertices is said to be pancyclic if it contains a cycle of length from 3 to n.

22 10 / 24 Pancyclicity We want to prove a similar theorem to the previous result for pancyclic, a stronger condition then hamiltonian. Definition: A graph G on n vertices is said to be pancyclic if it contains a cycle of length from 3 to n.

23 11 / 24 Metaconjecture: (Bondy, 1971) Almost any non-trivial condition on a graph G that guarantees G is hamiltonian also guarantees G is pancyclic possibly with a small number of exceptional graphs. Theorem: (Brandt, Favaron, Ryjá cek, 2000) For every k 2 there exists a k-connected claw-free graph that is not pancyclic.

24 12 / 24 Theorem: (Faudree, Gould, 1997) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 2-connected graph of order n 10. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. N(2, 0, 0) P 6

25 13 / 24 Theorem: (Gould, Łuczak, Pfender, 1997) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 3-connected graph. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. N(2, 1, 1) N(2, 2, 0) N(3, 1, 0) N(4, 0, 0) Ł P 7

26 14 / 24 Pancyclicity Theorem: (2012+) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 4-connected graph. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. Ł N(5, 1, 0) N(3, 3, 0) N(3, 2, 1) N(6, 0, 0) N(4, 2, 0) N(2, 2, 2) N(4, 1, 1) P 9

27 14 / 24 Pancyclicity Theorem: (2012+) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 4-connected graph. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. ( ) If G is claw-free and Y-free then G is pancylic. Gould, Łuczak, Pfender (1997) Ł

28 14 / 24 Pancyclicity Theorem: (2012+) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 4-connected graph. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. ( ) If G is claw-free and Y-free then G is pancylic. Ferrara, Gould, Genrke Magnant, Powell (2012) N(5, 1, 0) N(3, 3, 0) N(6, 0, 0) N(4, 2, 0)

29 14 / 24 Pancyclicity Theorem: (2012+) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 4-connected graph. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. ( ) If G is claw-free and Y-free then G is pancylic. Ferrara, Morris, Wenger (2012) P 9

30 14 / 24 Pancyclicity Theorem: (2012+) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 4-connected graph. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. ( ) If G is claw-free and Y-free then G is pancylic. C., Ferrara, Morris, Santana N(2, 2, 2) N(3, 2, 1) N(4, 1, 1)

31 15 / 24 Determine which subgraphs Y can possibly be. Two examples of claw-free nonpancyclic graphs. The graph Y must be an induced subgraph of both graphs.

32 15 / 24 Determine which subgraphs Y can possibly be. Two examples of claw-free nonpancyclic graphs. The graph Y must be an induced subgraph of both graphs.

33 16 / 24 Pancyclicity Possible candidates for Y. Known for P 9, N(, j, 0) where + j = 6. N(2, 2, 2) N(3, 2, 1) N(4, 1, 1)

34 17 / 24 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 5. Form G be replacing each vertex with e 2 e 2 e 1 e 3 e 1 K 12 e 3 e 4 e 4 G is claw-free, but not pancyclic.

35 18 / 24 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex with e

36 18 / 24 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex with e

37 18 / 24 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex with e

38 18 / 24 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex with e

39 18 / 24 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex with e

40 18 / 24 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex with e N(, j, k) + j + k = 6 P 9

41 19 / 24 Pancyclicity Possible candidates for Y. Known for P 9, N(, j, 0) where + j = 6. N(2, 2, 2) N(3, 2, 1) N(4, 1, 1)

42 19 / 24 Pancyclicity Possible candidates for Y. Known for P 9, N(, j, 0) where + j = 6. N(2, 2, 2) N(3, 2, 1) N(4, 1, 1)

43 20 / 24 If G is 4-connected, claw-free, and one of N(2, 2, 2), N(3, 2, 1) or N(4, 1, 1)-free then G is pancyclic. Build up Technique 1 Show G has cycles of small size. 2 Show that if G has an s-cycle, then G has an (s + 1)-cycle.

44 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

45 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

46 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

47 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

48 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

49 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

50 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

51 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

52 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

53 21 / 24 Since G is 4-connected, every vertex off the cycle has four vertex disjoint paths from to C. Pick such that the length of the three shortest paths is as small as possible. y z

54 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

55 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

56 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

57 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

58 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

59 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

60 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

61 22 / 24 Suppose G is claw-free and N(3, 2, 1)-free. Let C be an s-cycle of G. y z

62 23 / 24 Pancyclicity Theorem: (2012+) Let X and Y be connected graphs such that X, Y = P 3 and let G be a 4-connected graph. G is {X, Y}-free implies G is pancyclic if and only if X is the claw and Y is an induced subgraph of one of the following. Ł N(5, 1, 0) N(3, 3, 0) N(3, 2, 1) N(6, 0, 0) N(4, 2, 0) N(2, 2, 2) N(4, 1, 1) P 9

63 24 / 24 Pairs of Forbidden Subgraphs for Pancyclicity James Carraher University of Nebraska Lincoln s-jcarrah1@math.unl.edu Joint Work with Mike Ferrara, Tim Morris, Mike Santana September 2012