1 / 22 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
Hamiltonian 2 / 22 Definition: A hamiltonian cycle is a cycle that contains every vertex. A graph G is hamiltonian if it has a hamiltonian cycle.
Hamiltonian 2 / 22 Definition: A hamiltonian cycle is a cycle that contains every vertex. A graph G is hamiltonian if it has a hamiltonian cycle.
Hamiltonian 3 / 22 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(u) + d(v) n whenever uv / E(G), then G is hamiltonian. We are interested in conditions of connectivity and avoiding certain subgraphs implying the graph is hamiltonian.
4 / 22 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
4 / 22 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
5 / 22 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
5 / 22 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
5 / 22 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
5 / 22 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
5 / 22 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
6 / 22 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.
7 / 22 Generalized Net Hamiltonian We are interested in graphs that are claw-free that also avoid another subgraph. Definition: The generalized net N(i, j, k) is a triangle with three pendant paths of length i, j and k. N(2, 2, 2)
Hamiltonian 8 / 22 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
Hamiltonian 9 / 22 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.
10 / 22 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.
10 / 22 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.
10 / 22 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.
10 / 22 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.
10 / 22 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.
10 / 22 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.
Pancyclicity 11 / 22 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.
Pancyclicity 12 / 22 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
Pancyclicity 13 / 22 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
14 / 22 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
14 / 22 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
14 / 22 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)
14 / 22 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
14 / 22 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)
Pancyclicity 15 / 22 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.
Pancyclicity 15 / 22 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.
16 / 22 Pancyclicity Possible candidates for Y. Known for P 9, N(i, j, 0) where i + j = 6. N(2, 2, 2) N(3, 2, 1) N(4, 1, 1)
Pancyclicity 17 / 22 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 5. Form G be replacing each vertex v with e 2 e 2 e 1 e 3 v e 1 K 12 e 3 e 4 e 4 G is claw-free, but not pancyclic.
Pancyclicity 18 / 22 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex v with e
Pancyclicity 18 / 22 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex v with e
Pancyclicity 18 / 22 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex v with e
Pancyclicity 18 / 22 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex v with e
Pancyclicity 18 / 22 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex v with e
Pancyclicity 18 / 22 Example thanks to Kenta Ozeki. H is 4-connected, 4-regular, girth 7. Form G be replacing each vertex v with e N(i, j, k) i + j + k = 6 P 9
19 / 22 Pancyclicity Possible candidates for Y. Known for P 9, N(i, j, 0) where i + j = 6. N(2, 2, 2) N(3, 2, 1) N(4, 1, 1)
19 / 22 Pancyclicity Possible candidates for Y. Known for P 9, N(i, j, 0) where i + j = 6. N(2, 2, 2) N(3, 2, 1) N(4, 1, 1)
Pancyclicity 20 / 22 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.
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 21 / 22 Assume G has an s-cycle, C. Pick a vertex v off of C that minimizes three path lengths to the cycle. v
Pancyclicity 22 / 22 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