Spanning connectivity of graphs Elkin Vumar College of Mathematics - PowerPoint PPT Presentation

Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of graphs Elkin Vumar College of Mathematics and System Sciences Xinjiang University, P .R. China Shanghai Jiaotong University, China amss June 6, 2014 Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References Introduction 1 Some known results 2 Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs Some open problems and conjectures 3 Thanks 4 5 References amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References Spanning subgraph: A subgraph H of a graph G is spanning if V ( H ) = V ( G ) . Spanning paths and spanning cycles are widely known as Hamiltonian paths and Hamiltonian cycles, respectively. amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References Disjoint path covers in graphs Disjoint path cover Given an undirect graph G , a path cover is a set of paths in G where every vertex in G is covered by at least one path. A vertex-disjoint path cover, or simply disjoint path cover ( DPC for short) is the a path cover such that every vertex, possibly except for terminal vertices, belongs to one and only one path. Many-to-many k − DPC . Given two disjoint terminal sets S = { s 1 , s 2 , . . . , s k } and T = { t 1 , t 2 , . . . , t k } of G , each representing k sources and sinks, the many-to-many k − DPC is a DPC each of whose path joins a pair of source and sink. The DPC is regarded as paired if each source s i must be matched with a specified sink t i . The amss DPC is unpaired if any permutation of sinks may be mapped bijectively to sources. Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References Paired(unpaired) k -coverable graphs One-to-many k -DPC A one-to-many k -DPC is a k -DPC whose paths join a single source to k distinct sinks. One-to-one k -DPC A one-to-one k -DPC is a k -DPC whose paths join a single source to a single sink. Paired(unpaired) k -coverable graphs A graph G is called paired(resp. unpaired) k -coverable if | V ( G ) | ≥ 2 k and there always exists a paired (resp. unpaired) k -DPC for any S and T . amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References Spanning k -container C ( s , t ) or k ∗ -container: A one-to-one k -DPC between two vertices s and t is also called a spanning k -container C ( s , t ) or a k ∗ -container. Different terms Note that, for an integer k > 0 and s , t ∈ V ( G ) , s � = t , a spanning k -container C ( s , t ) ( k ∗ -container C ( s , t ) ) is also called “a spanning ( k , s , t ) -path-system" or “spanning k -rail between s and t " or “a spanning k -stave between s and t ". Spanning connectivity of graphs: A graph is k -spanning connected (or k ∗ -connected) if there is a k ∗ -container between any two distinct vertices. amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References Remark By definition, a graph G is 1 ∗ -connected if and only if it is Hamiltonian-connected, and G is 2 ∗ -connected if and only if it is Hamiltonian. All 1 ∗ -connected graphs except that K 1 and K 2 are 2 ∗ -connected. Hence spanning connectivity is a hybrid concept of Hamiltonicity and connectivity. amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of graphs: The spanning connectivity of of a graph G , denoted by κ ∗ ( G ) , is the largest integer k such that G is ω ∗ -connected for all 1 ≤ ω ≤ k . Note that κ ∗ ( G ) = 0 means that G is not Hamiltonian-connected. Super spanning connectedness: A graph G is super spanning connected if κ ∗ ( G ) = κ ( G ) . amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Spanning connectivity of general graphs Some open problems and conjectures Spanning connectivity of power of graphs Thanks Spanning connectivity index of graphs References Contents Introduction 1 Some known results 2 Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs Some open problems and conjectures 3 Thanks 4 References 5 amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Spanning connectivity of general graphs Some open problems and conjectures Spanning connectivity of power of graphs Thanks Spanning connectivity index of graphs References Dirac- and Ore-type conditions [C-K. Lin et al, 2008 DM] Assume that G is a graph with n ( G ) / 2 + 1 ≤ δ ( G ) ≤ n ( G ) − 2. Then G is r ∗ -connected for 1 ≤ r ≤ 2 δ ( G ) − n ( G ) + 2, where n ( G ) = | V ( G ) | and δ ( G ) is the minimum degree of G . [C-K. Lin et al, 2008 DM] Assume that k is a positive integer and there exists two nonadjacent vertices u and v with d ( u ) + d ( v ) ≥ n ( G ) + k . Then G is ( k + 2 ) ∗ -connected if and only G + uv is ( k + 2 ) ∗ -connected. Moreover, G is i ∗ -connected if and only if G + uv is i ∗ -connected for 1 ≤ i ≤ k + 2. [C-K. Lin et al, 2008 DM] Let k be a positive integer. If amss d ( u ) + d ( v ) ≥ n ( G ) + k for all nonadjacent vertices u and v of G , then G is r ∗ -connected for every 1 ≤ r ≤ k + 2. Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Spanning connectivity of general graphs Some open problems and conjectures Spanning connectivity of power of graphs Thanks Spanning connectivity index of graphs References Spanning connectivity of line graphs [Huang and Hsu,2011 AML] L ( G ) is k ∗ -connected if G is a multigraph with κ ′ ( G ) ≥ 2 k ≥ 4. Thus κ ∗ ( L ( G )) ≥ k if κ ′ ( G ) ≥ 2 k ≥ 4. [Huang and Hsu,2011 AML] L ( G ) is k ∗ -fan-connected if G is a multigraph with κ ′ ( G ) ≥ 2 k ≥ 4. Thus κ ∗ f ( L ( G )) ≥ k if κ ′ ( G ) ≥ 2 k ≥ 4. amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Spanning connectivity of general graphs Some open problems and conjectures Spanning connectivity of power of graphs Thanks Spanning connectivity index of graphs References Spanning connectivity and hamiltonian thickness of graphs and interval graphs [Peng Li and Yaokun Wu, 2014, DMTCS] Li and Wu discuss the spanning connectivity properties of graphs and interval graphs, and they discover that the existence of a thick Hamiltonian ordering guarantees that the graph has various kinds of spanning connectivity properties. For details of their work, see their paper entitled “Spanning connectivity and hamiltonian thickness of graphs and interval graphs" to appear in DMTCS. amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Spanning connectivity of general graphs Some open problems and conjectures Spanning connectivity of power of graphs Thanks Spanning connectivity index of graphs References Contents Introduction 1 Some known results 2 Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs Some open problems and conjectures 3 Thanks 4 References 5 amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Spanning connectivity of general graphs Some open problems and conjectures Spanning connectivity of power of graphs Thanks Spanning connectivity index of graphs References Pure bridge vertex and pure bridge triangle Pure bridge vertex A vertex of G is called a pure bridge vertex if each of its incident edges is a nontrivial bridge (neither of its two ends is of degree one). Pure bridge triangle A triangle with all vertices are of degree at least 3 is called a pure bridge triangle if every edge that incident with exactly endvertex of the triangle is a nontrivial bridge. Pure bridge pair A set of two adjacent vertices is called a pure bridge pair if both vertices are pure bridge vertices. amss Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Spanning connectivity of general graphs Some open problems and conjectures Spanning connectivity of power of graphs Thanks Spanning connectivity index of graphs References One-to-three 3-DPC in cubes of connected graphs Theorem (Park and Ihm, 2013, IPL) Let S = { s } and T = { t 1 , t 2 , t 3 } be subsets of V ( G ) . Then G 3 has a 3 ∗ -fan-container (or one-to-many 3-DPC) joining S and T if and only if (C1) and (C2) are satisfied: ( C 1 ) : There exists no pure bridge vertex v in G such that T ⊆ N G [ v ] and s / ∈ N G [ v ] ; ( C 2 ) : T does not form a pure bridge triangle in G such that s / ∈ N G [ T ] . amss Elkin Vumar Spanning connectivity of graphs