1. Mike Jacobson UCD Graphs that have Hamiltonian Cycles Avoiding Sets of Edges EXCILL November 20,2006

2. Mike Jacobson UCDHSC Graphs that have Hamiltonian Cycles Avoiding Sets of Edges EXCILL November 20,2006

3. Mike Jacobson UCDHSC-DDC Graphs that have Hamiltonian Cycles Avoiding Sets of Edges EXCILL November 20,2006

4. Part I - Containing There are many (MANY) results that give a condition for a graph (sufficiently large, bipartite, directed…) in order to assure that the graph contains ____________________ Recently (or NOT) there have been many (MANY) results presented that give a condition for a graph with (matching, 1-factor, disjoint cycles, “long cycles”, hamiltonian path or cycle, 2-factor, k-factor…) which contains some smaller predetermined substructure of the graph.

5. Specific Result Dirac Condition: If G is a graph with  ≥ (n+1)/2 and e is any edge of G, then G contains a hamiltonian cycle H containing e. So, (n+1)/2 is in fact necessary & best possible! K n/2,n/2 U tK 2

6. Another Example Ore Condition: If G is a graph with  2 ≥ n+1 and e is any edge of G, then G contains a hamiltonian cycle H containing e. Other Conditions – Number of Edges, high connectivity, Forbidden Subgraphs, neighborhood union, etc… This condition, n+1, is also best possible!!

7. More Examples - matchings t- matching in a k-matching (t < k) t- matching in a perfect-matching (t < n/2) t- matching on a hamiltonian path or cycle t- matching in a k-factor

8. More Examples – Linear Forests L(t, k) in a spanning linear forest L(t, k) in a spanning tree L(t, k) on a hamiltonian path or cycle L(t, k) on cycles of all possible lengths L(t, k) is a linear forest with t edges and k components L(t, k) in an r-factor L(t, k) in a 2-factor with k components

9. More Examples - digraphs arc - traceable arc - hamiltonian arc - pancyclic k – arc - …

10. More Examples – “Ordered” t- matching on a cycle in a specific order t- matching on a ham. cycle in a specific order t- matching on a cycle of all “possible” lengths in a specific order L(t,k) on a cycle of all possible lengths in a specific order

11. More Examples – “Equally Spaced” t- matching on a cycle (in a specific order) equally spaced around the cycle t- matching on a ham. cycle (in a specific order) equally spaced around the cycle t- matching on a cycle of all “possible” lengths (in a specific order) equally spaced around the cycle L(t,k) on a cycle of all “possible” lengths (in a specific order) equally spaced around the cycle

12. More Odds and Ends… putting vertices, edges, paths on different cycles in a set of disjoint cycles or 2-factor Hamiltonian cycle in a “larger” subgraph Many versions for bipartite graphs, hypergraphs… … Added conditions, connectivity, independence number, forbidden subgraphs…

13. If G is a bipartite graph of order n, with k ≥ 1, n ≥ 4k -2,  ≥ (n+1)/2 and v 1, v 2,..., v k distinct vertices of G then (1) G can be partitioned into k cycles C 1, C 2,..., C k such that v i is on C i for i = 1,..., k, or (2) k = 2 and G – {v 1, v 2 } = 2K (n-1)/2, (n-1)/2 and v2v2 v1v1 Claim 5.23 of Lemma 10 – when...

14. Part II - Avoiding (matching, 1-factor, disjoint cycles, “long cycles”, hamiltonian path or cycle, 2-factor, k-factor…) Preliminary Report!! which avoids every substructure of a particular type?? Are there any (ANY) results that give a condition for a graph (sufficiently large, bipartite, directed…) in order to assure that the graph contains ____________________ Joint with Mike Ferrara & Angela Harris

15. “Hamilton cycles, avoiding prescribed arcs, in close-to-regular tournaments” “Hamiltonian cycles avoiding prescribed arcs in tournaments” “Hamiltonian dicycles avoiding prescribed arcs in tournaments” There are some …

16. “Hamilton cycles, avoiding prescribed arcs, in close-to-regular tournaments” (1999) “Hamiltonian cycles avoiding prescribed arcs in tournaments” (1997) “Hamiltonian dicycles avoiding prescribed arcs in tournaments” (1987) There are some …

17. Results on Graphs and Bipartite Graphs Dirac, Ore and Moon & Moser – “conditions” Considering the problem for digraphs and tournaments

18. Ore Condition: If G is a graph with  2 ≥ n and e is any edge of G, then G contains a hamiltonian cycle H that avoids e?? Do we “get” anything for “free”?? K n-1 How large does  2 have to be??

19. Dirac Condition: If G is a graph with  ≥ n/2 and e is any edge of G, then G contains a hamiltonian cycle H that avoids e?? Do we “get” anything for “free”?? Dirac Condition: If G is a graph with  ≥ n/2 and E is any set of k edges of G, then G contains a hamiltonian cycle H that avoids E??

20. n/2 + 1 n/2 - 1 Add a (n+2)/4 - matching Let E be any subset of (n-2)/4 of the matching edges Theorem: If G is a graph of order n with  ≥ n/2 and E is any set of at most (n-6)/4 edges of G, then G contains a hamiltonian cycle H that avoids E. Note, that E is any set of (n-6)/4 edges n = 4k+2  ≥ n/2

21. Theorem: Let G be a graph with order n and H a graph of order at most n/2 and maximum degree k. If  2 ≥ n+k then G is H-avoiding hamiltonian. This is sharp for all choices of H With no restriction on the order of H…

22. Additional results on Bipartite Graphs Dirac, Ore and Moon & Moser – “conditions” Considering the problem for digraphs and tournaments We get results on extending any set of perfect matchings And on extending any set of hamiltonian cycles