The extendability of matchings in strongly regular graphs Sebastian - PowerPoint PPT Presentation

The extendability of matchings in strongly regular graphs Sebastian Cioab˘ a Weiqiang Li Department of Mathematical Sciences University of Delaware Villanova, June 5, 2014 Weiqiang Li The extendability of matchings in SRGs

Introduction Matching A set of edges M of a graph G is a matching if no two edges of M share a vertex. Weiqiang Li The extendability of matchings in SRGs

Introduction Matching A set of edges M of a graph G is a matching if no two edges of M share a vertex. Perfect Matching A matching M is perfect if every vertex is incident with exactly one edge of M . Weiqiang Li The extendability of matchings in SRGs

Example Weiqiang Li The extendability of matchings in SRGs

Example Weiqiang Li The extendability of matchings in SRGs

Extendability Extendability (Plummer (1980)) A graph G of even order v is called t -extendable if it contains at least one perfect matching, t < v / 2 and any matching of size t is contained in some perfect matching. The extendability of a graph G of even order is defined as the maximum t < v / 2 such that G is t -extendable Weiqiang Li The extendability of matchings in SRGs

Extendability Extendability (Plummer (1980)) A graph G of even order v is called t -extendable if it contains at least one perfect matching, t < v / 2 and any matching of size t is contained in some perfect matching. The extendability of a graph G of even order is defined as the maximum t < v / 2 such that G is t -extendable Problem Determine the extendability of a graph. Weiqiang Li The extendability of matchings in SRGs

Extendability Extendability (Plummer (1980)) A graph G of even order v is called t -extendable if it contains at least one perfect matching, t < v / 2 and any matching of size t is contained in some perfect matching. The extendability of a graph G of even order is defined as the maximum t < v / 2 such that G is t -extendable Problem Determine the extendability of a graph. Remark Zhang and Zhang (2006) obtained an O ( ve ) algorithm for determining the extendability of a bipartite graph G of order v and size e . At present time, the complexity of determining the extendability of a non-bipartite graph is unknown. Weiqiang Li The extendability of matchings in SRGs

1-extendable Graphs Theorem (Plesn´ ık) Let G be a ( k − 1) -edge-connected, k-regular graph with an even number of vertices. Then G is 1-extendable. Theorem (Lov´ asz and Plummer) Any vertex-transitive graph with even order is 1 -extendable. Weiqiang Li The extendability of matchings in SRGs

Strongly Regular Graphs Definition A graph G is a strongly regular graph with parameters v , k , λ and µ (shorthanded ( v , k , λ, µ )-SRG) if It has v vertices, is k -regular Any two adjacent vertices have exactly λ common neighbors Any two non-adjacent vertices have exactly µ common neighbors. (9,4,1,2)-SRG (10,3,0,1)-SRG Weiqiang Li The extendability of matchings in SRGs

Extendability of Strongly Regular Graph Theorem (Brouwer and Mesner (1985) ) If G is a primitive strongly regular graph of valency k, then G is k-connected. Any disconnecting set of size k must be the neighborhood of some vertex. Weiqiang Li The extendability of matchings in SRGs

Extendability of Strongly Regular Graph Theorem (Brouwer and Mesner (1985) ) If G is a primitive strongly regular graph of valency k, then G is k-connected. Any disconnecting set of size k must be the neighborhood of some vertex. Theorem (Lou and Zhu (1996)) Every connected strongly regular graph with even order is 2-extendable when k ≥ 3 , except the Petersen graph and the (6 , 4 , 2 , 4) -SRG graph. 2 1 1 3 4 2 5 6 3 5 8 7 10 4 6 9 Weiqiang Li The extendability of matchings in SRGs

Main tools Lemma (Yu (1993) ) Let t ≥ 1 and let G be a graph containing a perfect matching. The graph G is not t-extendable if and only if there exists a subset S ⊂ V ( G ) such that S contains t independent edges and o ( G − S ) ≥ | S | − 2 t + 2 . Weiqiang Li The extendability of matchings in SRGs

Main tools Lemma (Yu (1993) ) Let t ≥ 1 and let G be a graph containing a perfect matching. The graph G is not t-extendable if and only if there exists a subset S ⊂ V ( G ) such that S contains t independent edges and o ( G − S ) ≥ | S | − 2 t + 2 . Lemma (Cioab˘ a and Li (2014)) Let G be a strongly regular graph. If A is a subset of the vertex set such that 3 ≤ | A | ≤ v / 2 , then e ( A , A C ) ≥ 3 k − 6 Weiqiang Li The extendability of matchings in SRGs

Our results Theorem (Cioab˘ a and Li (2014)) Let G be a connected ( v , k , λ, µ ) -srg with v even and k ≥ 5 . Then G is 3 -extendable unless G is the complete 4 -partite graph K 2 , 2 , 2 , 2 (the (8 , 6 , 4 , 6) -srg), the complement of the Petersen graph (the (10 , 6 , 3 , 4) -srg) or the Shrikhande graph (one of the two (16 , 6 , 2 , 2) -srgs). Weiqiang Li The extendability of matchings in SRGs

Lower Bound for Extendability of SRGs: Dense Case Lemma If G is a primitive ( v , k , λ, µ ) -srg of even order with independence number α , then the extendability of G is � = ⌈ k 2 ⌉ If α = 2 − α ( G ) ≥ ⌈ k +3 2 ⌉ − 1 If α ≥ 3 2 Weiqiang Li The extendability of matchings in SRGs

Lower Bound for Extendability of SRGs: Dense Case Lemma If G is a primitive ( v , k , λ, µ ) -srg of even order with independence number α , then the extendability of G is � = ⌈ k 2 ⌉ If α = 2 − α ( G ) ≥ ⌈ k +3 2 ⌉ − 1 If α ≥ 3 2 The proof of this lemma uses the following fact. Theorem (Brouwer and Mesner (1985) ) If G is a primitive strongly regular graph of valency k, then G is k-connected. Any disconnecting set of size k must be the neighborhood of some vertex. Weiqiang Li The extendability of matchings in SRGs

Lower Bound for Extendability of SRGs: Dense Case Lemma If G is a primitive ( v , k , λ, µ ) -srg of even order with independence number α , then the extendability of G is � = ⌈ k 2 ⌉ − 1 If α = 2 − α ( G ) ≥ ⌈ k +3 2 ⌉ − 1 If α ≥ 3 2 Theorem If G is a ( v , k , λ, µ ) -srg with k / 2 < µ < k and α ≥ 3 , then the extendability of G is � � ⌈ k +3 − 3 k − 2 λ − 3 ≥ max 2(2 θ 2 +1) ⌉ − 1 , ⌈ λ/ 2 + 1 ⌉ 2 Weiqiang Li The extendability of matchings in SRGs

Lower Bound for Extendability of SRGs: Dense Case Lemma If G is a primitive ( v , k , λ, µ ) -srg of even order with independence number α , then the extendability of G is � = ⌈ k 2 ⌉ − 1 If α = 2 − α ( G ) ≥ ⌈ k +3 2 ⌉ − 1 If α ≥ 3 2 Theorem If G is a ( v , k , λ, µ ) -srg with k / 2 < µ < k and α ≥ 3 , then the extendability of G is � � ⌈ k +3 − 3 k − 2 λ − 3 ≥ ⌈ k +1 ≥ max 2(2 θ 2 +1) ⌉ − 1 , ⌈ λ/ 2 + 1 ⌉ 4 ⌉ . 2 Weiqiang Li The extendability of matchings in SRGs

Lower Bound for Extendability of SRGs: Sparse Case Theorem Let G be a ( v , k , λ, µ ) -srg with λ ≥ 1 . If µ ≤ k / 2 , then the extendability of G is � k 2 − k − 3 � � k � ≥ − 1 ≥ . 3 k − 7 3 Weiqiang Li The extendability of matchings in SRGs

Lower Bound for Extendability of SRGs: Sparse Case Theorem Let G be a ( v , k , λ, µ ) -srg with λ ≥ 1 . If µ ≤ k / 2 , then the extendability of G is � k 2 − k − 3 � � k � ≥ − 1 ≥ . 3 k − 7 3 Remark This result is close to being best possible as many strongly regular graph of valency k with λ ≥ 1 are not ⌈ k / 2 ⌉ -extendable. Weiqiang Li The extendability of matchings in SRGs

Classification of SRGs Theorem (Neumaier (1979)) Let m ≥ 2 be a fixed integer. Then with finitely many exceptions, the SRGs with smallest eigenvalue − m are of one of the following types: (a) Complete multipartite graphs with classes of size m, (b) Latin square graphs with parameters ( n 2 , m ( n − 1) , n − 2 + ( m − 1)( m − 2) , m ( m − 1)) , (c) Block graphs of Steiner m-systems with parameters � m − 1 , ( m − 1) 2 + n − 1 m − 1 − 2 , m 2 � m ( m − 1) , m ( n − m ) n ( n − 1) . Remark When m = 3, there are 66 other parameter sets. When m = 4, there are 232 other parameter sets. Weiqiang Li The extendability of matchings in SRGs

The exact extendability of some specific SRGs Theorem Let G be the line graph of K n , n with n ≥ 4 and n even. The extendability of G is k / 2 = n − 1 . Theorem Let T ( n ) be the triangular graph with parameters � n � ( , 2( n − 2) , n − 2 , 4) . If n ≥ 4 , the extendability of T ( n ) is 2 k / 2 − 1 = n − 3 . Weiqiang Li The extendability of matchings in SRGs

The exact extendability of some specific SRGs Theorem Let G be the block graph of a Steiner m-system on n points such m ( m − 1) is even. If m ∈ { 3 , 4 } and n > m 2 or m ≥ 5 and n ( n − 1) that n > 4 m 2 + 5 m + 24 + 96 m − 4 , the extendability of G is ⌈ k / 2 ⌉ − 1 , where k is the valency of G. Weiqiang Li The extendability of matchings in SRGs