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

Purpose of this study We give sharp degree sum conditions for partitioning graphs into a prescribed number of “chorded cycles”, and we show the di ff erence between cycles and chorded cycles in terms of sharp degree conditions. JCCA2018 Sendai 21 May 2018 01 / 10

G � graph of order n — — ∗ Hamilton cycle of G def. ⇒ cycle of G containing all vertices ⇐ Major study � Hamilton cycle 1 � “better” su ffi cient conditions 2 � “relaxed” structures JCCA2018 Sendai 21 May 2018 02 / 10

G � graph of order n — — ∗ Hamilton cycle of G def. ⇒ cycle of G containing all vertices ⇐ Major study � Hamilton cycle 1 � “better” degree conditions 2 � “relaxed” structures (Dirac 1952) n ≥ 3 � δ ( G ) ≥ n/ 2 ∃ Hamilton cycle ⇒ (Ore 1960) n ≥ 3 � σ 2 ( G ) ≥ n ∃ Hamilton cycle ⇒ ∗ δ ( G ) = min � d G ( x ) : x ∈ V ( G ) � � � ∗ σ 2 ( G ) = min d G ( x ) + d G ( y ) : x, y ∈ V ( G ) , xy / ∈ E ( G ) JCCA2018 Sendai ( � =) 21 May 2018 ( σ 2 ( G ) = + ∞ if G ： complete) degree of x 02 / 10

G � graph of order n — — ∗ Hamilton cycle of G def. ⇒ cycle of G containing all vertices ⇐ Major study � Hamilton cycle 1 � “better” degree conditions 2 � partition into cycles (Dirac 1952) n ≥ 3 � δ ( G ) ≥ n/ 2 ∃ Hamilton cycle ⇒ (Ore 1960) n ≥ 3 � σ 2 ( G ) ≥ n ∃ Hamilton cycle ⇒ ∗ δ ( G ) = min � d G ( x ) : x ∈ V ( G ) � � � ∗ σ 2 ( G ) = min d G ( x ) + d G ( y ) : x, y ∈ V ( G ) , xy / ∈ E ( G ) JCCA2018 Sendai ( � =) 21 May 2018 ( σ 2 ( G ) = + ∞ if G ： complete) degree of x 02 / 10

G � graph of order n , k ≥ 1 � int. — — ∗ Hamilton cycle of G C 1 C 2 def. ⇒ cycle of G containing all vertices ⇐ ∗ Partition of G into ( k ) cycles def. ( k ) disjoint cycles in G containing all vertices ⇐ ⇒ JCCA2018 Sendai 21 May 2018 03 / 10

G � graph of order n , k ≥ 1 � int. — — ∗ Hamilton cycle of G C 1 C 2 def. partition into 1 cycle ⇐ ⇒ ∗ Partition of G into ( k ) cycles def. ( k ) disjoint cycles in G containing all vertices ⇐ ⇒ JCCA2018 Sendai 21 May 2018 03 / 10

G � graph of order n , k ≥ 1 � int. — — ∗ Hamilton cycle of G C 1 C 2 def. partition into 1 cycle ⇐ ⇒ ∗ Partition of G into k cycles def. k disjoint cycles of G containing all vertices ⇐ ⇒ JCCA2018 Sendai 21 May 2018 03 / 10

G � graph of order n , k � 1 � int. — — ∗ Hamilton cycle of G C 1 C 2 def. partition into 1 cycle ⇐ ⇒ ∗ Partition of G into k cycles def. k disjoint cycles of G containing all vertices ⇐ ⇒ (Brandt et al. 1997) n ≥ 4 k − 1 � σ 2 ( G ) ≥ n ∃ partition into k cycles ⇒ JCCA2018 Sendai 21 May 2018 03 / 10

G � graph of order n , k ≥ 1 � int. — — ∗ Hamilton cycle of G C 1 C 2 def. partition into 1 cycle ⇐ ⇒ ∗ Partition of G into k cycles def. k disjoint cycles of G containing all vertices ⇐ ⇒ (Brandt et al. 1997) n ≥ 4 k − 1 � σ 2 ( G ) ≥ n ∃ partition into k cycles ⇒ Direction of study � partitions into k cycles with “additional conditions” ∗ containing pre-specified edges ∗ containing pre-specified vertices ∗ length constraints etc. JCCA2018 Sendai [CY] S. Chiba, T. Yamashita, Degree conditions for the existence of vertex-disjoint cycles 21 May 2018 and paths: a survey , Graphs Combin. 34 (2018) 1–83. 03 / 10

G � graph of order n , k ≥ 1 � int. — — ∗ Chorded cycle of G def. ⇒ subgraph of G consisting of ⇐ a cycle and edges joining two vertices of the cycle not cycle edges chord JCCA2018 Sendai 21 May 2018 04 / 10

G � graph of order n , k ≥ 1 � int. — — ∗ Chorded cycle of G def. ⇒ subgraph of G consisting of ⇐ a cycle and edges joining two vertices of the cycle not cycle edges chord (Remark) A chorded cycle is a relaxed structure of a complete graph = Chorded cycle of order t with t ( t − 3) Complete graph of order t chords 2 � of chords Many Few (Hamilton) cycle complete graph JCCA2018 Sendai 21 May 2018 04 / 10

G � graph of order n , k ≥ 1 � int. — — ∗ Chorded cycle of G def. ⇒ subgraph of G consisting of ⇐ a cycle and edges joining two vertices of the cycle not cycle edges chord (Remark) A chorded cycle is a relaxed structure of a complete graph = Chorded cycle of order t with t ( t − 3) Complete graph of order t chords 2 � of chords Many Few (Hamilton) cycle complete graph We first consider the following simple problem JCCA2018 Prob � Determine sharp degree conditions Sendai 21 May 2018 for partitioning graphs into k chorded cycles 04 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — JCCA2018 Sendai 21 May 2018 05 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — (Brandt et al. 1997) n ≥ 4 k − 1 � σ 2 ( G ) ≥ n ∃ partition into k cycles ⇒ (Qiao, Zhang 2012) Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G δ ( G ) ≥ n/ 2 ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ k cycles ( n − 1) / 2 vertices Exm.) (Remark) K ( n − 1) / 2 , ( n +1) / 2 δ condition is sharp ( n + 1) / 2 vertices JCCA2018 Sendai 21 May 2018 05 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — (Brandt et al. 1997) n ≥ 4 k − 1 � σ 2 ( G ) ≥ n ∃ partition into k cycles ⇒ (Qiao, Zhang 2012) Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G δ ( G ) ≥ n/ 2 ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ k cycles Which is better? deg. condition the rest conclusion Qiao, Zhang incomparable Brandt et al. JCCA2018 Sendai 21 May 2018 05 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — (Brandt et al. 1997) n ≥ 4 k − 1 � σ 2 ( G ) ≥ n ∃ partition into k cycles ⇒ (Qiao, Zhang 2012) Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G δ ( G ) ≥ n/ 2 ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ k cycles Which is better? deg. condition the rest conclusion Qiao, Zhang incomparable Brandt et al. Q1 � Can we improve the degree condition in Qiao-Zhang? JCCA2018 Sendai Q2 � Can we improve the conclusion in Brandt et al? 21 May 2018 05 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — Q1 � Can we improve the degree condition in Qiao-Zhang? Main Theorem 1 Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G σ 2 ( G ) ≥ n ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ (Qiao, Zhang 2012) Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G δ ( G ) ≥ n/ 2 ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ JCCA2018 Sendai 21 May 2018 06 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — Q1 � Can we improve the degree condition in Qiao-Zhang? Main Theorem 1 Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G σ 2 ( G ) ≥ n ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ Q2 � Can we improve the conclusion in Brandt et al? Main Theorem 2 n ≥ 4 k + 2 s − 1 � σ 2 ( G ) ≥ n ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ (Brandt et al. 1997) n ≥ 4 k − 1 � σ 2 ( G ) ≥ n ∃ partition into k cycles ⇒ JCCA2018 Sendai 21 May 2018 06 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — Q1 � Can we improve the degree condition in Qiao-Zhang? Main Theorem 1 Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G σ 2 ( G ) ≥ n ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ Q2 � Can we improve the conclusion in Brandt et al? Main Theorem 2 n ≥ 4 k + 2 s − 1 � σ 2 ( G ) ≥ n ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ Exm.) (Remark) K ( n − 1) / 2 , ( n +1) / 2 1 � σ 2 condition is sharp 2 � n ≥ 4 k + 2 s − 1 is necessary JCCA2018 Sendai 21 May 2018 06 / 10

G � graph of order n , k � 1, k � s � 0 � ints — — Q1 � Can we improve the degree condition in Qiao-Zhang? Main Theorem 1 Suppose ∃ s chorded cycles and ( k − s ) cycles, all disjoint, in G σ 2 ( G ) ≥ n ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ Q2 � Can we improve the conclusion in Brandt et al? Main Theorem 2 n ≥ 4 k + 2 s − 1 � σ 2 ( G ) ≥ n ∃ partition into s chorded cycles and ( k − s ) cycles ⇒ Exm.) (Remark) K 2 k + s − 1 , 2 k + s − 1 1 � σ 2 condition is sharp 2 � n ≥ 4 k + 2 s − 1 is necessary JCCA2018 Sendai 21 May 2018 06 / 10

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