Dirac-Erd˝ os Type Problems Kierstead-Kostochka-McConvey, 2016 (link) Let k ≥ 3 be an integer and G be a graph such that G does not contain two disjoint triangles. If V ≥ 2 k − V ≤ 2 k − 2 ≥ 2 k , then G contains k disjoint cycles. Question: do we really need to avoid disjoint triangles? Short answer: yes. Long answer: sometimes. Kierstead-Kostochka-McConvey, 2018 (link) Let k ≥ 2 be an integer and G be a graph with | G | ≥ 19 k and V ≥ 2 k − V ≤ 2 k − 2 ≥ 2 k . Then G contains k disjoint cycles. Open Characterize graphs G with V ≥ 2 k − V ≤ 2 k − 2 ≥ 2 k and no k disjoint cycles. 34 / 180
Outline Disjoint Cycles 1 Corr´ adi-Hajnal Tolerance for some low-degree vertices Ore condition (minimum degree-sum of nonadjacent vertices) Generalized Degree-Sum Conditions Connectivity Neighborhood Union Chorded Cycles 2 Degree conditions Neighborhood Union Multiply Chorded Cycles Equitable Coloring 3 Definition Connection to Cycles 35 / 180
Enomoto, Wang Corr´ adi-Hajnal, 1963 If G is a graph on n vertices with n ≥ 3 k and δ ( G ) ≥ 2 k , then G contains k disjoint cycles. 36 / 180
Enomoto, Wang Corr´ adi-Hajnal, 1963 If G is a graph on n vertices with n ≥ 3 k and δ ( G ) ≥ 2 k , then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } 37 / 180
Enomoto, Wang Corr´ adi-Hajnal, 1963 If G is a graph on n vertices with n ≥ 3 k and δ ( G ) ≥ 2 k , then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. 38 / 180
Enomoto, Wang Corr´ adi-Hajnal, 1963 If G is a graph on n vertices with n ≥ 3 k and δ ( G ) ≥ 2 k , then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Implies Corr´ adi-Hajnal 39 / 180
Enomoto, Wang Corr´ adi-Hajnal, 1963 If G is a graph on n vertices with n ≥ 3 k and δ ( G ) ≥ 2 k , then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Implies Corr´ adi-Hajnal Low degree vertices OK as long as they’re in a clique 40 / 180
Enomoto, Wang Corr´ adi-Hajnal, 1963 If G is a graph on n vertices with n ≥ 3 k and δ ( G ) ≥ 2 k , then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Implies Corr´ adi-Hajnal Low degree vertices OK as long as they’re in a clique With a little work, implies Dirac-Erd˝ os 41 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Proof (Enomoto) 42 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Proof (Enomoto) Edge-maximal counterexample 43 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Proof (Enomoto) Edge-maximal counterexample ◮ ( k − 1) disjoint cycles 44 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Proof (Enomoto) Edge-maximal counterexample ◮ ( k − 1) disjoint cycles ◮ Remaining graph at least 3 vertices 45 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Proof (Enomoto) Edge-maximal counterexample ◮ ( k − 1) disjoint cycles ◮ Remaining graph at least 3 vertices Minimize number of vertices in cycles 46 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Proof (Enomoto) Edge-maximal counterexample ◮ ( k − 1) disjoint cycles ◮ Remaining graph at least 3 vertices Minimize number of vertices in cycles Maximize longest path in remainder 47 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Sharpness: k k k 2 k − 1 48 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Sharpness: k k k 2 k − 1 3 k vertices 49 / 180
Enomoto, Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Sharpness: k k k 2 k − 1 3 k vertices α ( G ) large 50 / 180
Kierstead-Kostochka-Yeager, 2017 (link) Independence Number: Observation: α ( G ) ≥ n − 2 k + 1 ⇒ no k cycles 2 k − 1 51 / 180
Kierstead-Kostochka-Yeager, 2017 (link) Independence Number: Observation: α ( G ) ≥ n − 2 k + 1 ⇒ no k cycles Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. 52 / 180
Kierstead-Kostochka-Yeager, 2017 (link) Independence Number: Observation: α ( G ) ≥ n − 2 k + 1 ⇒ no k cycles Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. Kierstead-Kostochka-Yeager, 2017 (link) For k ≥ 4, if G is a graph on n vertices with n ≥ 3 k + 1 and σ 2 ( G ) ≥ 4 k − 3, then G contains k disjoint cycles if and only if α ( G ) ≤ n − 2 k . 53 / 180
Kierstead-Kostochka-Yeager, 2017 Kierstead-Kostochka-Yeager, 2017 (link) For k ≥ 4, if G is a graph on n vertices with n ≥ 3 k + 1 and σ 2 ( G ) ≥ 4 k − 3, then G contains k disjoint cycles if and only if α ( G ) ≤ n − 2 k . 54 / 180
Kierstead-Kostochka-Yeager, 2017 Kierstead-Kostochka-Yeager, 2017 (link) For k ≥ 4, if G is a graph on n vertices with n ≥ 3 k + 1 and σ 2 ( G ) ≥ 4 k − 3, then G contains k disjoint cycles if and only if α ( G ) ≤ n − 2 k . n ≥ 3 k + 1 k k k 2 k − 1 k 55 / 180
Kierstead-Kostochka-Yeager, 2017 Kierstead-Kostochka-Yeager, 2017 (link) For k ≥ 4, if G is a graph on n vertices with n ≥ 3 k + 1 and σ 2 ( G ) ≥ 4 k − 3, then G contains k disjoint cycles if and only if α ( G ) ≤ n − 2 k . k = 1: 56 / 180
Kierstead-Kostochka-Yeager, 2017 Kierstead-Kostochka-Yeager, 2017 (link) For k ≥ 4, if G is a graph on n vertices with n ≥ 3 k + 1 and σ 2 ( G ) ≥ 4 k − 3, then G contains k disjoint cycles if and only if α ( G ) ≤ n − 2 k . k = 2: u v 57 / 180
Kierstead-Kostochka-Yeager, 2017 Kierstead-Kostochka-Yeager, 2017 (link) For k ≥ 4, if G is a graph on n vertices with n ≥ 3 k + 1 and σ 2 ( G ) ≥ 4 k − 3, then G contains k disjoint cycles if and only if α ( G ) ≤ n − 2 k . k = 3: 58 / 180
Kierstead-Kostochka-Yeager, 2017 Kierstead-Kostochka-Yeager, 2017 (link) For k ≥ 4, if G is a graph on n vertices with n ≥ 3 k + 1 and σ 2 ( G ) ≥ 4 k − 3, then G contains k disjoint cycles if and only if α ( G ) ≤ n − 2 k . σ 2 = 4 k − 4: k − 3 2 r K 2 t k + 1 k + 3 2 r − 2 59 / 180
Outline Disjoint Cycles 1 Corr´ adi-Hajnal Tolerance for some low-degree vertices Ore condition (minimum degree-sum of nonadjacent vertices) Generalized Degree-Sum Conditions Connectivity Neighborhood Union Chorded Cycles 2 Degree conditions Neighborhood Union Multiply Chorded Cycles Equitable Coloring 3 Definition Connection to Cycles 60 / 180
Extending Enomoto-Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } 61 / 180
Extending Enomoto-Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } � � � σ t ( G ) = min d ( V ) : I is an independent set of size t v ∈ I 62 / 180
Extending Enomoto-Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } � � � σ t ( G ) = min d ( V ) : I is an independent set of size t v ∈ I Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. 63 / 180
Extending Enomoto-Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } � � � σ t ( G ) = min d ( V ) : I is an independent set of size t v ∈ I Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. t = 1: Corr´ adi-Hajnal t = 2: Enomoto-Wang 64 / 180
Extending Enomoto-Wang Enomoto 1998, Wang 1999 If G is a graph on n vertices with n ≥ 3 k and σ 2 ( G ) ≥ 4 k − 1, then G contains k disjoint cycles. σ 2 ( G ) := min { d ( x ) + d ( y ) : xy �∈ E ( G ) } � � � σ t ( G ) = min d ( V ) : I is an independent set of size t v ∈ I Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. t = 1: Corr´ adi-Hajnal t = 2: Enomoto-Wang t = 3: Fujita, Matsumura, Tsugaki, Yamashita 2006 (link) t = 4: proved in paper as evidence for conjecture 65 / 180
Ma, Yan Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. True for t ≤ 4. 66 / 180
Ma, Yan Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. True for t ≤ 4. Ma, Yan 2018+ (link) Let G be a graph with | G | ≥ (2 t + 1) k . If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 5, then G contains k disjoint cycles. 67 / 180
Ma, Yan Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. True for t ≤ 4. Ma, Yan 2018+ (link) Let G be a graph with | G | ≥ (2 t + 1) k . If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 5, then G contains k disjoint cycles. Proof In an edge-maximal counterexample, choose k − 1 disjoint cycles such that number of vertices in cycles is minimal, and number of connected components in remaining graph is minimal 68 / 180
Ma, Yan Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. True for t ≤ 4. Ma, Yan 2018+ (link) Let G be a graph with | G | ≥ (2 t + 1) k . If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 5, then G contains k disjoint cycles. Degree-sum condition is sharp: 69 / 180
Ma, Yan Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. True for t ≤ 4. Ma, Yan 2018+ (link) Let G be a graph with | G | ≥ (2 t + 1) k . If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 5, then G contains k disjoint cycles. Degree-sum condition is sharp: 2 k − 1 70 / 180
Ma, Yan Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. True for t ≤ 4. Ma, Yan 2018+ (link) Let G be a graph with | G | ≥ (2 t + 1) k . If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 5, then G contains k disjoint cycles. Degree-sum condition is sharp: 2 k − 1 71 / 180
Ma, Yan Conjecture: Gould, Hirohata, Keller 2018 (link) Let G be a graph of sufficiently large order. If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 1, then G contains k disjoint cycles. True for t ≤ 4. Ma, Yan 2018+ (link) Let G be a graph with | G | ≥ (2 t + 1) k . If σ t ( G ) ≥ 2 kt − t + 1 for any two integers k ≥ 2 and t ≥ 5, then G contains k disjoint cycles. Open What is the best possible bound on | G | in the Ma-Yan Theorem? Can we characterize graphs G with σ t ( G ) ≥ 2 kt − t + 1 but no k disjoint cycles? 72 / 180
Outline Disjoint Cycles 1 Corr´ adi-Hajnal Tolerance for some low-degree vertices Ore condition (minimum degree-sum of nonadjacent vertices) Generalized Degree-Sum Conditions Connectivity Neighborhood Union Chorded Cycles 2 Degree conditions Neighborhood Union Multiply Chorded Cycles Equitable Coloring 3 Definition Connection to Cycles 73 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected graphs do not have k disjoint cycles? 74 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected graphs do not have k disjoint cycles? Observation: G is (2 k − 1) connected 75 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected graphs do not have k disjoint cycles? Observation: G is (2 k − 1) connected ⇒ δ ( G ) ≥ 2 k − 1 76 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected graphs do not have k disjoint cycles? Observation: G is (2 k − 1) connected ⇒ δ ( G ) ≥ 2 k − 1 ⇒ σ 2 ( G ) ≥ 4 k − 2 77 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected graphs do not have k disjoint cycles? Observation: G is (2 k − 1) connected ⇒ δ ( G ) ≥ 2 k − 1 ⇒ σ 2 ( G ) ≥ 4 k − 2 KKY: Holds for σ 2 ( G ) ≥ 4 k − 3 78 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected graphs do not have k disjoint cycles? Answer to Dirac’s Question for Simple Graphs (KKY 2017) Let k ≥ 2. Every graph G with ( i ) | G | ≥ 3 k and ( ii ) δ ( G ) ≥ 2 k − 1 contains k disjoint cycles if and only if if k is odd and | G | = 3 k , then G � = 2 K k ∨ K k , and α ( G ) ≤ | G | − 2 k , and if k = 2 then G is not a wheel. k k k 2 k − 1 79 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected graphs do not have k disjoint cycles? Answer to Dirac’s Question for Simple Graphs (KKY 2017) Let k ≥ 2. Every graph G with ( i ) | G | ≥ 3 k and ( ii ) δ ( G ) ≥ 2 k − 1 contains k disjoint cycles if and only if if k is odd and | G | = 3 k , then G � = 2 K k ∨ K k , and α ( G ) ≤ | G | − 2 k , and if k = 2 then G is not a wheel. Further: characterization for multigraphs 80 / 180
Simple Graphs → Multigraphs Idea: Take all 1-vertex cycles 81 / 180
Simple Graphs → Multigraphs Idea: Take all 1-vertex cycles 82 / 180
Simple Graphs → Multigraphs Idea: Take all 1-vertex cycles Take as many 2-vertex cycles as possible (maximum matching) 83 / 180
Simple Graphs → Multigraphs Idea: Take all 1-vertex cycles Take as many 2-vertex cycles as possible (maximum matching) 84 / 180
Simple Graphs → Multigraphs Idea: Take all 1-vertex cycles Take as many 2-vertex cycles as possible (maximum matching) What’s left is a simple graph 85 / 180
(2 k − 1)-connected multigraphs with no k disjoint cycles Answer to Dirac’s Question for multigraphs: Kierstead-Kostochka-Yeager 2015 (link) Let k ≥ 2 and n ≥ k . Let G be an n -vertex graph with simple degree at least 2 k − 1 and no loops. Let F be the simple graph induced by the strong edgs of G , α ′ = α ′ ( F ), and k ′ = k − α ′ . Then G does not contain k disjoint cycles if and only if one of the following holds: n + α ′ < 3 k ; | F | = 2 α ′ (i.e., F has a perfect matching) and either (i) k ′ is odd and G − F = Y k ′ , k ′ , or (ii) k ′ = 2 < k and G − F is a wheel with 5 spokes; G is extremal and either (i) some big set is not incident to any strong edge, or (ii) for some two distinct big sets I j and I j ′ , all strong edges intersecting I j ∪ I j ′ have a common vertex outside of I j ∪ I j ′ ; n = 2 α ′ + 3 k ′ , k ′ is odd, and F has a superstar S = { v 0 , . . . , v s } with center v 0 such that either (i) G − ( F − S + v 0 ) = Y k ′ +1 , k ′ , or (ii) s = 2, v 1 v 2 ∈ E ( G ), G − F = Y k ′ − 1 , k ′ and G has no edges between { v 1 , v 2 } and the set X 0 in G − F ; k = 2 and G is a wheel, where some spokes could be strong edges; k ′ = 2, | F | = 2 α ′ + 1 = n − 5, and G − F = C 5 . 86 / 180
k ′ odd, F has a perfect matching Example: k = 8, α ′ = 3, k ′ = 5. k ′ k ′ k ′ 87 / 180
Big independent set, incident to no multiple edges 2 k − 1 88 / 180
Wheel, with possibly some spokes multiple Example: k = 2 89 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected multigraphs do not have k disjoint cycles? Kierstead-Kostochka-Yeager 2015 (link) Characterization of multigraphs without k disjoint cycles that have minimum simple degree at least 2 k − 1. That is, the underlying simple graph G has δ ( G ) ≥ 2 k − 1. 90 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected multigraphs do not have k disjoint cycles? Kierstead-Kostochka-Yeager 2015 (link) Characterization of multigraphs without k disjoint cycles that have minimum simple degree at least 2 k − 1. That is, the underlying simple graph G has δ ( G ) ≥ 2 k − 1. Kierstead-Kostochka-Molla-Yager 2018+ (link) Characterization of multigraphs without k disjoint cycles that have minimum simple degree sum of nonadjacent vertices at least 4 k − 3. That is, the underlying simple graph G has σ 2 ( G ) ≥ 4 k − 3. 91 / 180
Dirac: (2 k − 1)-connected without k disjoint cycles Dirac, 1963 (link) What (2 k − 1)-connected multigraphs do not have k disjoint cycles? Kierstead-Kostochka-Yeager 2015 (link) Characterization of multigraphs without k disjoint cycles that have minimum simple degree at least 2 k − 1. That is, the underlying simple graph G has δ ( G ) ≥ 2 k − 1. Kierstead-Kostochka-Molla-Yager 2018+ (link) Characterization of multigraphs without k disjoint cycles that have minimum simple degree sum of nonadjacent vertices at least 4 k − 3. That is, the underlying simple graph G has σ 2 ( G ) ≥ 4 k − 3. Open Do the other results in this talk generalize nicely to multigraphs? 92 / 180
Outline Disjoint Cycles 1 Corr´ adi-Hajnal Tolerance for some low-degree vertices Ore condition (minimum degree-sum of nonadjacent vertices) Generalized Degree-Sum Conditions Connectivity Neighborhood Union Chorded Cycles 2 Degree conditions Neighborhood Union Multiply Chorded Cycles Equitable Coloring 3 Definition Connection to Cycles 93 / 180
Neighborhood Union Faudree-Gould, 2005 (link) If G has n ≥ 3 k vertices and | N ( x ) ∪ N ( y ) | ≥ 3 k for all nonadjacent pairs of vertices x , y , then G contains k disjoint cycles. 94 / 180
Neighborhood Union Faudree-Gould, 2005 (link) If G has n ≥ 3 k vertices and | N ( x ) ∪ N ( y ) | ≥ 3 k for all nonadjacent pairs of vertices x , y , then G contains k disjoint cycles. x y 95 / 180
Neighborhood Union Faudree-Gould, 2005 (link) If G has n ≥ 3 k vertices and | N ( x ) ∪ N ( y ) | ≥ 3 k for all nonadjacent pairs of vertices x , y , then G contains k disjoint cycles. x y d ( x ) + d ( y ) = 6 96 / 180
Neighborhood Union Faudree-Gould, 2005 (link) If G has n ≥ 3 k vertices and | N ( x ) ∪ N ( y ) | ≥ 3 k for all nonadjacent pairs of vertices x , y , then G contains k disjoint cycles. x y d ( x ) + d ( y ) = 6 | N ( x ) ∪ N ( y ) | = 4 97 / 180
Neighborhood Union Faudree-Gould, 2005 (link) If G has n ≥ 3 k vertices and | N ( x ) ∪ N ( y ) | ≥ 3 k for all nonadjacent pairs of vertices x , y , then G contains k disjoint cycles. Neither stronger nor weaker than Corr´ adi-Hajnal. If δ ( G ) = 2 k , then xy �∈ E ( G ) {| N ( x ) ∪ N ( y ) |} ≥ 2 k . min If | N ( x ) ∪ N ( y ) | ≥ 3 k , then δ ( G ) ≥ 0. 98 / 180
Neighborhood Union Faudree-Gould, 2005 (link) If G has n ≥ 3 k vertices and | N ( x ) ∪ N ( y ) | ≥ 3 k for all nonadjacent pairs of vertices x , y , then G contains k disjoint cycles. Proof In an edge-maximal counterexample, choose k − 1 disjoint cycles such that number of vertices in cycles is minimal, and number of connected components in remaining graph is minimal 99 / 180
Neighborhood Union Faudree-Gould, 2005 (link) If G has n ≥ 3 k vertices and | N ( x ) ∪ N ( y ) | ≥ 3 k for all nonadjacent pairs of vertices x , y , then G contains k disjoint cycles. Sharpness: K 3 k − 4 K 5 100 / 180
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