The Ryjacek Closure

and

a Forbidden Subgraph

Akira Saito (Nihon University, Japan)

Liming Xiong (Beijing Institute of Technology, China)

Ryjacek Closure

(Preaching Buddhism to Buddha)

In 1997, Ryjacek discovered a new closure.

Ryjacek closure

• a locally-connected vertex• local completion

A vertex in a graph is said to be locally-connected if (the

neighborhood of ) induces a connected graph in .

Local completion at is the operation of joining every pair of non-

adjacent neighbors of by an edge.

A vertex in a graph is said to be eligible if is locally-connected

and does not induce a complete graph.

Ryjacek Closure

: a sequence of graphs

•

• is obtained from by local completion at an eligible vertex

in .

: a graph

If does not contain an eligible vertex, we call a (the) Ryjacek

closure and denote it by clR.

appears to depend on the order of eligible vertices chosen for

local completion at each step.

Ryjacek Closure

Theorem A （ Ryjacek 1997 ）

1. The Ryjacek closure is uniquely determined, regardless of the order of

eligible vertices chosen for local completion.

2. If is a claw-free graph, then clR is a line graph.

3. If is a claw-free graph, then is hamiltonian if and only if clR is

hamiltonian.

(Note that the uniqueness of the Ryjacek closure is not limited to the

class of claw-free graphs.)

Ryjacek Closure

The Ryjacek closure is a powerful operation in the class of claw-free graphs.

Conjecture 1 (Matthews-Sumner Conjecture, 1984)

Every 4-connected claw-free graph is hamiltonian.

Conjecture 2 (Thomassen’s Conjecture, 1986)

Every 4-connected line graph is hamiltonian.

Corollary B (Ryjacek, 1997)

Conjecture 2 imlpies Conjecture 1.

Standard Approach

claw-free graph

line graph

pre-image

Ryjacek closure

Harary & Nash-Williams

super eulerian graphs and other techniques

Motivation

The Ryjacek closure is a powerful operation in the class of claw-free

graphs.

Is there any other class of graphs defined by a forbidden subgraph, in

which the Ryjacek closure works effectively?

: a connected graph of order at least three

𝑆 (𝐻 )=¿ “For every -free graph of sufficiently large order, is hamiltonian if

and only if clR is hamiltonian.”

Want : a graph which makes the statement true

Candidates

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

Proof (easy)

“For every -free graph of sufficiently large order, is hamiltonian if

and only if clR is hamiltonian.”

If holds, then a sufficiently large locally-connected -free graph is

hamiltonian.

Theorem 1

For a connected graph of order at least three, if the statment holds,

then H is , , , or .

(A graph is locally-connected if every vertex in is locally-connected.)

Proof (easy)

• non-hamiltonian

If holds, then a sufficiently large locally-connected -free graph is

hamiltonian.

• locally-connected

• can be arbitrarily large

This graph cannot be -free.

is an induced subgraph of this graph.

Proof (easy)

If holds, then a sufficiently large locally-connected -free graph is

hamiltonian.

𝐾 𝑠

Replace each edge in with the graph

on the right.

• non-hamiltonian

• locally-connected

• can be arbitrarily large

This graph cannot be -free.

is an induced subgraph of this graph.

Proof (easy)

is an induced subgraph of these graphs.

𝐾 𝑠

Replace each edge in with the graph

on the right.

H is , , , or .

Candidates

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

Check three graphs

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1Ryjacek’s result

Check three graphs

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

Check

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

Check

𝐾 1,2

“For every -free graph of sufficiently large order, is hamiltonian if

and only if clR is hamiltonian.”

What is a -free graph?

No pair of vertices of distance 2

Every component is a complete graph.

No eligible vertex

clR

is true, but trivial.

Check

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

Check

“For every -free graph of sufficiently large order, is hamiltonian if

and only if clR is hamiltonian.”

What is a -free graph?

The neighborhood of a vertex has no edge.

If is locally-connected, .

No eligible vertex

clR

is true, but trivial.

𝐾 3

Check

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

Check

“For every -free graph of sufficiently large order, is

hamiltonian if and only if clR is hamiltonian.”

What is a -free graph?

A graph is -free if and only if the neighborhood

of every vertex induces a -free graph

If is locally-connected, then induces a complete graph.

No eligible vertex clR

is true, but trivial.

𝐾 2+2𝐾 1

Check three graphs

Theorem 1

For a connected graph of order at least three, if the statement holds,

then H is , , , or .

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

These graphs make trivial.

Conclusion

𝐾 1,2 𝐾 3 𝐾 1 ,3 𝐾 2+2𝐾 1

𝑆 (𝐻 )=¿ “For every -free graph of sufficiently large order, is hamiltonian if

and only if clR is hamiltonian.”

Theorem 1

For a connected graph of order at least three, the statement holds if

and only if H is , , , or .

However, only gives a nontrivial statement.

Conclusion

𝑆 (𝐻 )=¿ “For every -free graph of sufficiently large order, is hamiltonian if

and only if clR is Hamiltonian.”

Theorem 1

For a connected graph of order at least three, the statement holds if

and only if H is , , , or .

The Ryjacek closure works in a non-trivial manner only in the class of

claw-free graphs.

Question

• The Ryjacek closure works in a non-trivial manner only in the class of

claw-free graphs.

But why is the Ryjacek closure so powerful in the class of claw-

free graphs, while it does now work effectively for other forbidden

subgraphs?

Is there any mechanism which makes the combination of the Ryjacek

closure and claw-free graphs so powerful?

• The Ryjacek closure works in a non-trivial manner only in the class of

claw-free graphs as long as we forbid one graph.

What happens if we forbid two or more graphs?

Děkuji mockrát !Thank you!