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

amss 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 June 6, 2014

Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

1

Introduction

2

Some known results Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

3

Some open problems and conjectures

4

Thanks

5

References

Elkin Vumar Spanning connectivity of graphs

amss 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.

Elkin Vumar Spanning connectivity of graphs

amss 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,

• r 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 = {s1, s2, . . . , sk} and T = {t1, t2, . . . , tk} 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 si must be matched with a specified sink ti. The DPC is unpaired if any permutation of sinks may be mapped bijectively to sources.

Elkin Vumar Spanning connectivity of graphs

amss 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)| ≥ 2k and there always exists a paired (resp. unpaired) k-DPC for any S and T.

Elkin Vumar Spanning connectivity of graphs

amss 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"

• r “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.

Elkin Vumar Spanning connectivity of graphs

amss 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 K1 and K2 are

2∗-connected. Hence spanning connectivity is a hybrid concept

• f Hamiltonicity and connectivity.

Elkin Vumar Spanning connectivity of graphs

amss 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).

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Contents

1

Introduction

2

Some known results Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

3

Some open problems and conjectures

4

Thanks

5

References

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

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 ≤ 2δ(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

• nly 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 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

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Spanning connectivity of line graphs

[Huang and Hsu,2011 AML] L(G) is k∗-connected if G is a multigraph with κ′(G) ≥ 2k ≥ 4. Thus κ∗(L(G)) ≥ k if κ′(G) ≥ 2k ≥ 4. [Huang and Hsu,2011 AML] L(G) is k∗-fan-connected if G is a multigraph with κ′(G) ≥ 2k ≥ 4. Thus κ∗

f (L(G)) ≥ k if

κ′(G) ≥ 2k ≥ 4.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

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.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Contents

1

Introduction

2

Some known results Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

3

Some open problems and conjectures

4

Thanks

5

References

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

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.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

One-to-three 3-DPC in cubes of connected graphs

Theorem (Park and Ihm, 2013, IPL) Let S = {s} and T = {t1, t2, t3} be subsets of V(G). Then G3 has a 3∗-fan-container (or one-to-many 3-DPC) joining S and T if and only if (C1) and (C2) are satisfied: (C1): There exists no pure bridge vertex v in G such that T ⊆ NG[v] and s / ∈ NG[v]; (C2): T does not form a pure bridge triangle in G such that s / ∈ NG[T].

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Spanning connectivity of Gk with k ≥ 3.

Theorem (Park and Ihm, 2013, IPL) For every connected graph G with at least 4 vertices, G3 is

• ne-to-one 3-coverable. (G3 is 3∗-connected for a connected

graph with at least 4 vertices.) Theorem (Sabir and Vumar, 2014,GC) For every connected graph G with at least k + 1 ≥ 4 vertices, Gk is one-to-one k-coverable. (If G is a connected graph of

• rder |V(G)| ≥ k + 1 ≥ 4, then Gk is k∗-connected.)

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Problem Determine the minimum number f = f(k, κ, d) such that the f-th power Gf of a graph G with connectivity κ ≥ 1 and diameter d is k∗-connected. In particular, it is of great interest to generalize the result of Fleischner, i.e., prove or disprove that G2 is 3∗-connected for a 2-connected graph of order at least 4.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Conjecture [Park and Ihm, 2013, IPL]The square of a 2-connected graph is

• ne-to-many 3-coverable.

Problem Find the necessary and sufficient condition under which G3 is

• ne-to-many k-coverable, where G is a connected graph of
• rder at least k + 1 ≥ 5.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Paired 2-DPC in cubes of connected graphs

Theorem (Park and Ihm, 2014, DM) Let S = {s1, s2} and T = {t1, t2} be terminal sets of a connected graph G. The cube G3 has a paired 2-DPC joining S and T if and only if (C1): s1, s2, t1, t2 / ∈ NG[v] for any pure bridge vertex v of G, and (C2): si, ti is not a pure bridge pair of G such that {s3−i, t3−i} ⊆ NG({si, ti}) for eachi = 1, 2.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Unpaired 2-DPC in cubes of connected graphs

Theorem (Park and Ihm, 2014, DM) Let S = {s1, s2} and T = {t1, t2} be terminal sets of a connected graph G. The cube G3 has an unpaired 2-DPC joining S and T if and only if {s1, s2, t1, t2} NG[v] for any pure bridge vertex v of G.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Conjecture (Park and Ihm, 2014, DM) The square of every 2-connected graph with at least 4 vertices is unpaired 2-coverable. Conjecture (Park and Ihm, 2014, DM) Let S = {s1, s2} and T = {t1, t2} be terminal sets of a 2-connected graph G with at least 4 vertices. The square G2 has no paired 2-DPC joining S and T if and only if (a) G is isomorphic to an even cycle (v0, v1, . . . , v|V(G)|−1), and (b) S and T are such that {s1, t1} = {v0, vq} and {s2, t2} = {vp, vr} for some even integers p, q and r with 0 < p < q < r.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Contents

1

Introduction

2

Some known results Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

3

Some open problems and conjectures

4

Thanks

5

References

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

P index of graphs

Iterated line graphs Let L0(G) = G and L1(G) = L(G)(the line graph of G), and Lk+1(G) = L(Lk(G)) for k ≥ 0. P-index of G Let P be a graph property. For a connected graph G such that G / ∈ {Pn, Cn, K1,3}, it is proved that the (edge-) connectivity of Lm(G) increases quickly with the increase of m. Hence it is of significance to study the least integer m for which Lm(G) has the property P, and this least integer is called the P-index of G.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Hamiltonian and Hamilton-connected indices of graphs

Hamiltonian index h(G) For a connected graph G that is not a path, the hamiltonian index h(G), is defined as h(G) = min{m : Lm(G) is Hamiltonian}. Hamilton-connected index hc(G) For a connected graph G such that G / ∈ {Pn, Cn, K1,3}, the Hamilton-connected index hc(G), is defined as hc(G) = min{m : Lm(G) is Hamilton − connected}. Spanning k-connected index of G For a connected graph G such that G / ∈ {Pn, Cn, K1,3}, the spanning k-connected index sk(G), is define sk(G) = min{m : Lm(G) is k∗ − connected}. Obviously, hc(G) = s1(G) and h(G) = s2(G).

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Special paths in graphs

A poly-vertex and poly-path in G A poly-vertex is one having degree at least 3, and a poly-path is a path joining two poly-vertices of G, whose internal vertices, if any, have degree 2. An end-path An end-path is a path joining poly-vertex with an end-vertex (leaf) of G, whose internal vertices, if any, have degree 2. A cyclic-poly-path A cyclic-poly-path is a path joining two poly-vertice and all internal vertices are on a cycle and of degree 2.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Some known results on h(G)

[Chartrand and Wall] If T is a tree which is not a path, then h(T) = max{{l(P) + 1}, {l(Q)}} over all poly-path P and end-path Q of T, where l(·) is the length of a path. [Catlin et al., and Lai] Let G be a graph that is neither a path nor C2. Let k be the longest lane in G. Then h(G) ≤ min{diam(G), k + 1}. [Xiong] Let G be a connected graph other than a path. Then h(G) ≤ diam(G) − 1.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Some known results on hc(G)

[Chen et al.] Let G be a connected graph that is neither a path nor Cn. If the length of a longest lane is k, then k − 1 ≤ hc(G) ≤ max{diam(G), k − 1}. [Chen et al.] Let G be a graph that is neither a path nor Cn. Then κ3G) ≤ hc(G) ≤ κ3(G) + 2, where κ3(G) = min{m : Lm(G) is 3 − connected}.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

An upper bound of s3(G)

[A divalent path in G] A (u, v)-path in G is called unclosed if u = v. An unclosed path in G is called a divalent path, if all its internal vertices are of degree 2 in G. [l(G)] Let l(G) be the length of a longest divalent path that is not a length 2 path of K3. [W. Xiong et al. 2014,J.C.O] Let G be a connected graph such that G / ∈ {Pn, Cn, K1,3}. Then s3(G) ≤ l(G) + 6.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Hamilton-connected index of trees and unicyclic graphs

[Sabir and Vumar, 2014,GC] If G is a tree with |V(T)| ≥ 5 which is not a path, then h(T) ≤ hc(T) ≤ h(T) + 1, where h(T) = max{{l(P) + 1}.{l(Q)}} over all poly-paths P and end-paths Q of T, and l(·) is the length of a path. [Sabir and Vumar, 2014,GC] Let G be a unicyclic graph with |V(G)| ≥ 4 that is not a cycle. Then k ≤ hc(G) ≤ max{k + 1, k′ + 1}, where k = max{{l(P) + 1}, {l(Q)}} over all poly-paths P and end-paths Q of G, l(·) is the length of a path, and k′ is the length of a longest cyclic-poly-path in G.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Hamilton-connected index of a special class of graphs

[Chartrand and Wall, 1973] If G is a connected graph containing cyclic and acyclic blocks such that each cyclic block is hamiltonian, then h(G) = max{{l(P) + 1}.{l(Q)}}

• ver all poly-paths P and end-paths Q of G, where l(·) is

the length of a path. [Sabir and Vumar, 2014,GC] If G is a connected graph containing cyclic and acyclic blocks such that each cyclic block is hamiltonian, then k ≤ hc(G) ≤ max{k + 1, k′ + 1}, where k = max{{l(P) + 1}, {l(Q)}} over all poly-paths P and end-paths Q, l(·) is the length of a path, and k′ is the length of a longest cyclic-poly-path in G.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Trees with hc(T) = h(T) + 1

[Sabir and Vumar, 2014,GC] Let T be a tree of order at least 5 that is not a path. Then hc(T) = h(T) + 1 if and only if one of the following holds: (i) h(T) = max{{l(P) + 1}, {l(Q)}} = l(P) + 1; (ii) h(T) = max{{l(P) + 1}, {l(Q)}} = l(Q); where P is a longest poly-path in T with ends u and v such that dT(u) ≥ dT(v) = 3 and Q is a longest end-path in T.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Unicyclic graphs with hc(G) = h(G) + 1

[Sabir and Vumar, 2014,GC] Let G be a unicyclic graph of order at least 4 that is not a cycle. Then hc(G) = h(G) + 1 if and only if one of the following holds: (i) h(G) = max{{l(P) + 1}, {l(Q)}} = l(P) + 1; (ii) h(G) = max{{l(P) + 1}, {l(Q)}} = l(Q); where P is a longest poly-path in G with ends u and v such that dG(u) ≥ dG(v) = 3 and Q is a longest end-path in G.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References Spanning connectivity of general graphs Spanning connectivity of power of graphs Spanning connectivity index of graphs

Specail class of graphs with hc(G) = h(G) + 1

[Sabir and Vumar, 2014,GC] Let G be a connected graph containing cyclic and acyclic blocks such that each cyclic block is hamiltonian. Then hc(G) = h(G) + 1 if and only if one of the following holds: (i) h(G) = max{{l(P) + 1}, {l(Q)}} = l(P) + 1; (ii) h(G) = max{{l(P) + 1}, {l(Q)}} = l(Q); where P is a longest poly-path in G with ends u and v such that dG(u) ≥ dG(v) = 3 and Q is a longest end-path in G.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

Conjecture 1

[P .Li and Y.Wu,2014, DMTCS] Let k be a positive integer. (i) If G is spanning (k + 1)-fan-connected, then it must be spanning k-fan-connected. (ii) If G is spanning (k + 2)-rail-connected, then it must be spanning (k + 1)-rail-connected.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

Problem 1

[Vumar and B. Wu, 2014] For given positive integers n and k ≥ 3, let f(n, k) = min{e(G) : |V(G)| = n and G is k∗ − connected}. Determine f(n, k) or find nontrivial bounds for f(n, k). In particular, determine f(n, 3).

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

Problem 2

[Vumar and B. Wu, 2014] For given positive integers n and k ≥ 3, let g(n, k) = min{e(G) : |V(G)| = n and κ∗(G) = κ(G)}. Determine g(n, k) or find nontrivial bounds for g(n, k). In particular, determine g(n, 3).

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

Problem 3

[Vumar, 2014] For a positive integer k ≥ 3 and a connected graph G such that G / ∈ {Pn, Cn, K1,3}, find the relationship between sk(G) and sk−1(G), sk−2(G), · · · . In particular, find the relationship between s3(G) and s1(G) = hc(G), s2(G) = h(G), maybe for some special classes of graphs.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

Thank you very much for your attention.

Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

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Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

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Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

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amss Introduction Some known results Some open problems and conjectures Thanks References

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amss Introduction Some known results Some open problems and conjectures Thanks References

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Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

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amss Introduction Some known results Some open problems and conjectures Thanks References

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Elkin Vumar Spanning connectivity of graphs

amss Introduction Some known results Some open problems and conjectures Thanks References

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Elkin Vumar Spanning connectivity of graphs

Introduction Some known results Some open problems and conjectures Thanks References

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