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