mike jacobson ucd graphs that have hamiltonian cycles avoiding sets of edges excill november 20,2006
TRANSCRIPT
Mike JacobsonUCD
Graphs that have Hamiltonian Cycles Avoiding Sets of Edges
EXCILLNovember 20,2006
Mike JacobsonUCDHSC
Graphs that have Hamiltonian Cycles Avoiding Sets of Edges
EXCILLNovember 20,2006
Mike JacobsonUCDHSC-DDC
Graphs that have Hamiltonian Cycles Avoiding Sets of Edges
EXCILLNovember 20,2006
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…)
(matching, 1-factor, disjoint cycles, “long cycles”, hamiltonian path or cycle, 2-factor, k-factor…)
which contains some smaller predetermined substructure of the graph.
Specific Result
Dirac Condition: If G is a graph with ≥ (n+1)/2
and e is any edge of G, then G contains ahamiltonian cycle H containing e.
So, (n+1)/2 is in fact necessary & best possible!
Kn/2,n/2
U tK2
Another Example
Ore Condition: If G is a graph with2 ≥ n+1
and e is any edge of G, then G contains ahamiltonian cycle H containing e.
Other Conditions – Number of Edges, high connectivity, Forbidden Subgraphs,
neighborhood union, etc…
This condition, n+1, is also best possible!!
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
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
More Examples - digraphs
arc - traceable
arc - hamiltonian
arc - pancyclic
k – arc - …
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” lengthsin a specific order
L(t,k) on a cycle of all possible lengthsin a specific order
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
cycleL(t,k) on a cycle of all “possible” lengths
(in a specific order) equally spaced around the cycle
More Odds and Ends…
putting vertices, edges, paths on different cyclesin a set of disjoint cycles or 2-factor
Hamiltonian cycle in a “larger” subgraph
Many versions for bipartite graphs,hypergraphs…
…
Added conditions, connectivity, independencenumber, forbidden subgraphs…
If G is a bipartite graph of order n, with k ≥ 1, n ≥ 4k -2, ≥ (n+1)/2 and v1, v2, . . . , vk distinct vertices
of G then
(1) G can be partitioned into k cycles C1, C2, . . . , Ck such that vi is on Ci for i = 1, . . . , k, or
(2) k = 2 and G – {v1, v2} = 2K(n-1)/2, (n-1)/2 and
v2
v1
Claim 5.23 of Lemma 10 – when . . .
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
“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 …
“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 …
Results on Graphs and Bipartite Graphs
Dirac, Ore and Moon & Moser – “conditions”
Considering the problem for digraphs and tournaments
Ore Condition: If G is a graph with2 ≥ n
and e is any edge of G, then G contains ahamiltonian cycle H that avoids e??
Do we “get” anything for “free”??
Kn-1
How large does 2 have to be??
Dirac Condition: If G is a graph with≥ n/2
and e is any edge of G, then G contains ahamiltonian 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??
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/2and 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
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
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…
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