Degree conditions for partitioning graphs into chorded cycles Shuya - - PowerPoint PPT Presentation

Degree conditions for partitioning graphs into chorded cycles

Shuya Chiba (Kumamoto University, Japan)

joint work with Shoichi Kamada (Kumamoto University, Japan)

JCCA 2018 Sendai International Center 21 May, 2018

JCCA2018 Sendai 21 May 2018

Purpose of this study

01 / 10

We give sharp degree sum conditions for partitioning graphs into a prescribed number of “chorded cycles”, and we show the difference between cycles and chorded cycles in terms of sharp degree conditions.

JCCA2018 Sendai 21 May 2018

def.

⇐ ⇒ cycle of G containing all vertices ∗ Hamilton cycle of G Major study 1“better” sufficient conditions 2“relaxed” structures

Hamilton cycle

— — Ggraph of order n

02 / 10

JCCA2018 Sendai 21 May 2018

def.

⇐ ⇒ cycle of G containing all vertices ∗ Hamilton cycle of G

(=)

(σ2(G) = +∞ if G：complete)

degree of x

∗ σ2(G) = min

• dG(x) + dG(y) : x, y ∈ V (G), xy /

∈ E(G)

• ∗ δ(G) = min dG(x) : x ∈ V (G)

Major study 1“better” 2“relaxed” structures

Hamilton cycle

— — Ggraph of order n degree conditions

02 / 10 (Ore 1960)

σ2(G) ≥ n

⇒ ∃Hamilton cycle ⇒ ∃Hamilton cycle

(Dirac 1952)

δ(G) ≥ n/2 n ≥ 3 n ≥ 3

JCCA2018 Sendai 21 May 2018

02 / 10

def.

⇐ ⇒ cycle of G containing all vertices ∗ Hamilton cycle of G

(Ore 1960)

σ2(G) ≥ n

⇒ ∃Hamilton cycle

(=)

(σ2(G) = +∞ if G：complete)

∗ σ2(G) = min

• dG(x) + dG(y) : x, y ∈ V (G), xy /

∈ E(G)

• ⇒

∃Hamilton cycle

(Dirac 1952)

∗ δ(G) = min dG(x) : x ∈ V (G) Major study 2

Hamilton cycle

— — Ggraph of order n

δ(G) ≥ n/2

1“better” degree conditions partition into cycles

degree of x

n ≥ 3 n ≥ 3

JCCA2018 Sendai 21 May 2018

def.

⇐ ⇒ C1 C2 ∗ Partition of G into (k) cycles — — Ggraph of order n, k ≥ 1int.

def.

⇐ ⇒ cycle of G containing all vertices ∗ Hamilton cycle of G

03 / 10

(k) disjoint cycles in G containing all vertices

JCCA2018 Sendai 21 May 2018

def.

⇐ ⇒ C1 C2 — — Ggraph of order n, k ≥ 1int.

03 / 10

def.

⇐ ⇒ ∗ Hamilton cycle of G partition into 1 cycle ∗ Partition of G into (k) cycles (k) disjoint cycles in G containing all vertices

JCCA2018 Sendai 21 May 2018

def.

⇐ ⇒ C1 C2 — — Ggraph of order n, k ≥ 1int.

03 / 10

def.

⇐ ⇒ ∗ Hamilton cycle of G partition into 1 cycle ∗ Partition of G into k cycles k disjoint cycles of G containing all vertices

JCCA2018 Sendai 21 May 2018

C1 C2 — —

Ggraph of order n, k 1int.

(Brandt et al. 1997)

⇒ σ2(G) ≥ n ∃partition into k cycles n ≥ 4k − 1

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def.

⇐ ⇒ ∗ Hamilton cycle of G partition into 1 cycle

def.

⇐ ⇒ ∗ Partition of G into k cycles k disjoint cycles of G containing all vertices

JCCA2018 Sendai 21 May 2018

[CY]

• S. Chiba, T. Yamashita, Degree conditions for the existence of vertex-disjoint cycles

and paths: a survey, Graphs Combin. 34 (2018) 1–83.

C1 C2

etc. ∗ containing pre-specified edges ∗ containing pre-specified vertices ∗ length constraints partitions into k cycles with “additional conditions” — — Ggraph of order n, k ≥ 1int.

(Brandt et al. 1997)

⇒ σ2(G) ≥ n ∃partition into k cycles n ≥ 4k − 1

def.

⇐ ⇒ ∗ Hamilton cycle of G Direction of study

03 / 10

partition into 1 cycle

def.

⇐ ⇒ ∗ Partition of G into k cycles k disjoint cycles of G containing all vertices

JCCA2018 Sendai 21 May 2018

∗ Chorded cycle of G

def.

⇐ ⇒ subgraph of G consisting of a cycle and edges joining two vertices of the cycle — — Ggraph of order n, k ≥ 1int.

not cycle edges chord

04 / 10

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∗ Chorded cycle of G

def.

⇐ ⇒ subgraph of G consisting of a cycle and edges joining two vertices of the cycle A chorded cycle is a relaxed structure of a complete graph

Few Many

• f chords

(Hamilton) cycle complete graph

— — Ggraph of order n, k ≥ 1int.

(Remark)

not cycle edges

04 / 10

= Chorded cycle of order t with t(t−3)

2

chords Complete graph of order t

chord

JCCA2018 Sendai 21 May 2018

04 / 10

∗ Chorded cycle of G

def.

⇐ ⇒ subgraph of G consisting of a cycle and edges joining two vertices of the cycle A chorded cycle is a relaxed structure of a complete graph

Few Many

• f chords

(Hamilton) cycle complete graph

Prob Determine sharp degree conditions We first consider the following simple problem — —

Ggraph of order n, k ≥ 1int.

(Remark)

not cycle edges

for partitioning graphs into k chorded cycles = Chorded cycle of order t with t(t−3)

2

chords Complete graph of order t

chord

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints

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JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints

(Brandt et al. 1997) (Qiao, Zhang 2012)

∃partition into s chorded cycles and (k − s) cycles

δ(G) ≥ n/2 ⇒

k cycles

Exm.) K(n−1)/2,(n+1)/2

(n − 1)/2 vertices (n + 1)/2 vertices

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G ⇒ σ2(G) ≥ n

∃partition into k cycles n ≥ 4k − 1

(Remark)

δ condition is sharp

05 / 10

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints

(Brandt et al. 1997) (Qiao, Zhang 2012)

∃partition into s chorded cycles and (k − s) cycles

δ(G) ≥ n/2 ⇒

k cycles

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G ⇒ σ2(G) ≥ n

∃partition into k cycles n ≥ 4k − 1

Which is better? conclusion

• deg. condition

the rest

Qiao, Zhang Brandt et al. incomparable

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JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints

(Brandt et al. 1997) (Qiao, Zhang 2012)

∃partition into s chorded cycles and (k − s) cycles

δ(G) ≥ n/2 ⇒

k cycles

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G ⇒ σ2(G) ≥ n

∃partition into k cycles n ≥ 4k − 1

Which is better? conclusion

• deg. condition

the rest

Brandt et al. Qiao, Zhang incomparable Q1Can we improve the degree condition in Qiao-Zhang? Q2Can we improve the conclusion in Brandt et al?

05 / 10

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang?

(Qiao, Zhang 2012)

∃partition into s chorded cycles and (k − s) cycles

δ(G) ≥ n/2 ⇒ Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

06 / 10

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang? Q2Can we improve the conclusion in Brandt et al?

⇒

Main Theorem 2

n ≥ 4k + 2s − 1σ2(G) ≥ n

∃partition into s chorded cycles and (k − s) cycles

(Brandt et al. 1997)

⇒ σ2(G) ≥ n ∃partition into k cycles n ≥ 4k − 1

06 / 10

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang? Q2Can we improve the conclusion in Brandt et al?

⇒

Main Theorem 2

n ≥ 4k + 2s − 1σ2(G) ≥ n

∃partition into s chorded cycles and (k − s) cycles Exm.) K(n−1)/2,(n+1)/2

06 / 10 (Remark)

1σ2 condition is sharp 2n ≥ 4k + 2s − 1 is necessary

JCCA2018 Sendai 21 May 2018

06 / 10

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang? Q2Can we improve the conclusion in Brandt et al?

⇒

Main Theorem 2

n ≥ 4k + 2s − 1σ2(G) ≥ n

∃partition into s chorded cycles and (k − s) cycles Exm.) K2k+s−1,2k+s−1

(Remark)

1σ2 condition is sharp 2n ≥ 4k + 2s − 1 is necessary

JCCA2018 Sendai 21 May 2018

07 / 10

— — Ggraph of order n, k 1, k s 0ints Q1Can we improve the degree condition in Qiao-Zhang? Q2Can we improve the conclusion in Brandt et al?

⇒

Main Theorem 2

n ≥ 4k + 2s − 1σ2(G) ≥ n

∃partition into s chorded cycles and (k − s) cycles

(Remark)

⇒

(Chiba, Fujita, Gao, Li 2010)

n ≥ 3k + sσ2(G) ≥ 4k + 2s − 1

(∗)

• Thm. 2 is obtained from Thm. 1 and the following theorem

∃partition into s chorded cycles and (k − s) cycles

⇒

(∗)

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

• σ2(G) ≥ n

Main Theorem 1

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

(∗)

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

• σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang?

1 to the case of chorded cycles

(even the case of chorded cycles with c chords)

We can use “crossing arguments” in hamiltonian problems

08 / 10

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

(∗)

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

• σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang?

1 H x u

if ∃

H Ci

C1, . . . , Ckmax. k disjoint

Ci

• Hcomp. of G − Ci

successor of u

u v chorded cycles, to the case of chorded cycles

(even the case of chorded cycles with c chords)

We can use “crossing arguments” in hamiltonian problems

08 / 10

JCCA2018 Sendai 21 May 2018

08 / 10

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

(∗)

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

• σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang?

1 to the case of chorded cycles

(even the case of chorded cycles with c chords)

H x u

successor of u if ∃

H u Ci v Ci

C1, . . . , Ckmax. k disjoint

• Hcomp. of G − Ci

chorded cycles, We can use “crossing arguments” in hamiltonian problems C

i

C

i

JCCA2018 Sendai 21 May 2018

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

(∗)

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

• σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang?

Ci 2 to the case of chorded cycles H Cj u v (if ∃“good interval”)

C1, . . . , Ckmax. k disjoint

• Hcomp. of G − Ci

chorded cycles, We can use “insertion arguments” in k cycles partition problems

09 / 10

JCCA2018 Sendai 21 May 2018

09 / 10

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

(∗)

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

• σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang?

Ci 2 to the case of chorded cycles H Cj u v (if ∃“good interval”)

This is also a chord of C

i

We can use “insertion arguments” in k cycles partition problems C

j

C

i

C1, . . . , Ckmax. k disjoint

• Hcomp. of G − Ci

chorded cycles,

JCCA2018 Sendai 21 May 2018

09 / 10

— — Ggraph of order n, k 1, k s 0ints ∃partition into s chorded cycles and (k − s) cycles

⇒

(∗)

Suppose ∃s chorded cycles and (k − s) cycles, all disjoint, in G

• σ2(G) ≥ n

Main Theorem 1

Q1Can we improve the degree condition in Qiao-Zhang?

Ci 2 to the case of chorded cycles H Cj u v (if ∃“good interval”)

This is also a chord of C

i

We can use “insertion arguments” in k cycles partition problems C

j

C

i

C1, . . . , Ckmax. k disjoint

• Hcomp. of G − Ci

chorded cycles,

JCCA2018 Sendai 21 May 2018

Prob.

Determine sharp deg conditions for partitioning graphs into k chorded cycles with c chords σ2 n

• f chords

deg conditions

σ2 n 1

2 c

• Few

Many

Brandt et al. Main Thms

10 / 10

JCCA2018 Sendai 21 May 2018

Prob.

Determine sharp deg conditions for partitioning graphs into k chorded cycles with c chords σ2 n

• f chords

deg conditions

σ2 n 1

2 c

• Few

Many

Brandt et al. Main Thms

degree conditions for ∃k disjoint chorded cycles with c chords

Known

⇒

(∗)

(Chen et al. 2015)

n k,cδ(G)

• c + 1 + 1k + 12 · (9/2)c

∃k disjoint chorded cycles with c chords

(it may not be a partition) (σ2 condition depending on only k, c also implies )

(∗)

(Chiba, Lichiardopol 2018)

10 / 10

JCCA2018 Sendai 21 May 2018

10 / 10 Prob.

Determine sharp deg conditions for partitioning graphs into k chorded cycles with c chords σ2 n

• f chords

deg conditions

σ2 n 1

2 c

• Few

Many

Brandt et al. Main Thms

degree conditions for ∃k disjoint chorded cycles with c chords

Known

⇒

(∗)

(Chen et al. 2015)

n k,cδ(G)

• c + 1 + 1k + 12 · (9/2)c

∃k disjoint chorded cycles with c chords

(it may not be a partition) (σ2 condition depending on only k, c also implies )

(∗)

(Chiba, Lichiardopol 2018)

Q This also implies the following? The k disjoint chorded cycles can be transformed a partition of G

⇒ (∗)

n k, c, δ(G) n/2 (or σ2(G) ≥ n)

Thank you for your attention!