Post on 20-Jan-2016
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