Mike Jacobson UCD

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

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

3. Graphs that have Hamiltonian Cycles Avoiding Sets of Edges Mike Jacobson UCDHSC-DDC 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 ____________________ (matching, 1-factor, disjoint cycles, “long cycles”, hamiltonian path or cycle, 2-factor, k-factor…) 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 d≥ (n+1)/2 and e is any edge of G, then G contains a hamiltonian cycle H containing e. U tK2 Kn/2,n/2 So, (n+1)/2 is in fact necessary & best possible!

6. Another Example Ore Condition: If G is a graph with s2≥ n+1 and e is any edge of G, then G contains a hamiltonian cycle H containing e. This condition, n+1, is also best possible!! Other Conditions – Number of Edges, high connectivity, Forbidden Subgraphs, neighborhood union, etc…

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) is a linear forest with t edges and k components 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) in a 2-factor with k components L(t, k) in an r-factor

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, d≥ (n+1)/2 and v1, v2, . . . , vk distinct vertices of G then • G can be partitioned into k cycles C1, C2, . . . , Ck such that vi is on Ci for i = 1, . . . , k, or • k = 2 and G – {v1, v2} = 2K(n-1)/2, (n-1)/2 and v1 v2 Claim 5.23 of Lemma 10 – when . . .

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

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

16. There are some … “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)

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

18. Do we “get” anything for “free”?? Ore Condition: If G is a graph with s2≥ n and e is any edge of G, then G contains a hamiltonian cycle H that avoids e?? Kn-1 How large does s2 have to be??

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

20. n = 4k+2 n/2 + 1 n/2 - 1 Add a (n+2)/4 - matching d ≥ n/2 Let E be any subset of (n-2)/4 of the matching edges Theorem: If G is a graph of order n with d ≥ 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

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 s2≥ n+k then G is H-avoiding hamiltonian. This is sharp for all choices of H With no restriction on the order of H… Theorem: Let G be a graph with order n and H a graph of order at most n/2 and maximum degree k. If s2≥ n+k then G is H-avoiding hamiltonian. This is sharp for all choices of H

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