On s -fully cycle extendable line graphs Yehong Shao Ohio - - PowerPoint PPT Presentation

On s-fully cycle extendable line graphs

Yehong Shao Ohio University Southern, Ironton, OH 45638

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Definitions

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fully cycle extendable √ A graph G is said to be fully cycle extendable if every vertex of G lies on a triangle and for every nonhamiltonian cycle C there is a cycle C′ in G such that V (C) ⊆ V (C′) and |V (C′)| = |V (C)| + 1.

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fully cycle extendable √ A graph G is said to be fully cycle extendable if every vertex of G lies on a triangle and for every nonhamiltonian cycle C there is a cycle C′ in G such that V (C) ⊆ V (C′) and |V (C′)| = |V (C)| + 1. √ If the removal of any s vertices in G results in a fully cycle extendable graph, we say G is an s-fully cycle extendable graph.

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Line Graphs

√ L(G): the line graph of a graph G, has E(G) as its vertex set, where two vertices in L(G) are linked if and only if the corresponding edges in G share a common vertex.

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Line Graphs

√ L(G): the line graph of a graph G, has E(G) as its vertex set, where two vertices in L(G) are linked if and only if the corresponding edges in G share a common vertex. √ If G is simple, L(G) is also simple.

t t t t t t

G

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Line Graphs

√ L(G): the line graph of a graph G, has E(G) as its vertex set, where two vertices in L(G) are linked if and only if the corresponding edges in G share a common vertex. √ If G is simple, L(G) is also simple.

t t t t t t ❞ ❞ ❞ ❞ ❞

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Line Graphs

√ L(G): the line graph of a graph G, has E(G) as its vertex set, where two vertices in L(G) are linked if and only if the corresponding edges in G share a common vertex. √ If G is simple, L(G) is also simple.

t t t t t t ❞ ❞ ❞ ❞ ❞

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Line Graphs

√ L(G): the line graph of a graph G, has E(G) as its vertex set, where two vertices in L(G) are linked if and only if the corresponding edges in G share a common vertex. √ If G is simple, L(G) is also simple.

t t t t t t

G: solid lines and closed circles L(G): dash lines and open circles

❞ ❞ ❞ ❞ ❞

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Iterated Line Graphs

√ For a nontrivial connected graph G, we define L0(G) = G and for any integer k > 0, Lk(G) = L(Lk−1(G)).

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Iterated Line Graphs

√ For a nontrivial connected graph G, we define L0(G) = G and for any integer k > 0, Lk(G) = L(Lk−1(G)). √ G

q q q q q

v0 v1 v2

❅ ❅

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Iterated Line Graphs

√ For a nontrivial connected graph G, we define L0(G) = G and for any integer k > 0, Lk(G) = L(Lk−1(G)). √ G

q q q q q

v0 v1 v2

❅ ❅

• √ L(G)

q q q q ❅ ❅

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Iterated Line Graphs

√ For a nontrivial connected graph G, we define L0(G) = G and for any integer k > 0, Lk(G) = L(Lk−1(G)). √ G

q q q q q

v0 v1 v2

❅ ❅

• √ L(G)

q q q q ❅ ❅

• √ L2(G)

q q q q ❅ ❅

• ❅

❅

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Early Studies of Iterated Line graphs

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Early Results √ Let G be a graph.

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Early Results √ Let G be a graph. √ Minimum degree=: δ(G); Maximum degree=: ∆(G); Edge-connectivity =: κ′(G); Vertex-connectivity=: κ(G).

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Early Results √ Let G be a graph. √ Minimum degree=: δ(G); Maximum degree=: ∆(G); Edge-connectivity =: κ′(G); Vertex-connectivity=: κ(G). √ (Chartrand and Stewart, 1969) If G is k-connected, then Li(G) is [2i−1(k − 2) + 2]-connected and [2i(k − 2) + 2]-edge-connected.

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Early Results √ (Chartrand and Stewart, 1969) If G is k-connected, then Li(G) is [2i−1(k − 2) + 2]-connected and [2i(k − 2) + 2]-edge-connected.

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Early Results √ (Chartrand and Stewart, 1969) If G is k-connected, then Li(G) is [2i−1(k − 2) + 2]-connected and [2i(k − 2) + 2]-edge-connected. √ (Niepel, Knor, and Šolte´ s, 1996) Conjecture For any connected graph G that is not a path, there exists an integer K such that, for all i ≥ K, ∆(Li+1(G)) = 2∆(Li(G)) − 2.

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Early Results √ (Chartrand and Stewart, 1969) If G is k-connected, then Li(G) is [2i−1(k − 2) + 2]-connected and [2i(k − 2) + 2]-edge-connected. √ (Niepel, Knor, and Šolte´ s, 1996) Conjecture For any connected graph G that is not a path, there exists an integer K such that, for all i ≥ K, ∆(Li+1(G)) = 2∆(Li(G)) − 2. √ (Niepel, Knor, and Šolte´ s, 1996) Conjecture For any connected graph G that is not a path, there exists an integer K such that, for all i ≥ K, δ(Li+1(G)) = 2δ(Li(G)) − 2.

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Early Results √ (Chartrand and Stewart, 1969) If G is k-connected, then Li(G) is [2i−1(k − 2) + 2]-connected and [2i(k − 2) + 2]-edge-connected. √ (Niepel, Knor, and Šolte´ s, 1996) Conjecture For any connected graph G that is not a path, there exists an integer K such that, for all i ≥ K, ∆(Li+1(G)) = 2∆(Li(G)) − 2. √ (Niepel, Knor, and Šolte´ s, 1996) Conjecture For any connected graph G that is not a path, there exists an integer K such that, for all i ≥ K, δ(Li+1(G)) = 2δ(Li(G)) − 2. √ Hartke and Higgins proved both conjectures using induced subgraphs of maximum in 1999(minimum in 2003) degree vertices and locally maximum (minimum) vertices.

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Hamiltonian Properties of Iterated Line Graphs √ In 1973, Chartrand introduced the hamiltonian index of a connected graph G that is not a path to be the minimum number of applications of the line graph operator so that the resulting graph is hamiltonian. He showed that the hamiltonian index exists as a Þnite number.

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Hamiltonian Properties of Iterated Line Graphs √ In 1973, Chartrand introduced the hamiltonian index of a connected graph G that is not a path to be the minimum number of applications of the line graph operator so that the resulting graph is hamiltonian. He showed that the hamiltonian index exists as a Þnite number. √ In 1983, Clark and Wormald extended this idea of Chartrand and introduced the hamiltonian-like indices.

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Hamiltonian Properties of Iterated Line Graphs √ In 1973, Chartrand introduced the hamiltonian index of a connected graph G that is not a path to be the minimum number of applications of the line graph operator so that the resulting graph is hamiltonian. He showed that the hamiltonian index exists as a Þnite number. √ In 1983, Clark and Wormald extended this idea of Chartrand and introduced the hamiltonian-like indices. √ The hamiltonian index, h(G), of a connected graph G is the least nonnegative integer m such that Lm(G) is Hamiltonian.

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Hamiltonian Properties of Iterated Line Graphs √ In 1973, Chartrand introduced the hamiltonian index of a connected graph G that is not a path to be the minimum number of applications of the line graph operator so that the resulting graph is hamiltonian. He showed that the hamiltonian index exists as a Þnite number. √ In 1983, Clark and Wormald extended this idea of Chartrand and introduced the hamiltonian-like indices. √ The hamiltonian index, h(G), of a connected graph G is the least nonnegative integer m such that Lm(G) is Hamiltonian. √ The s-hamiltonian index, hs(G), of a connected graph G is the least nonnegative integer m such that Lm(G) is s-Hamiltonian.

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Hamiltonian Properties of Iterated Line Graphs √ Define l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path in G is a path in G whose interval vertices have degree two in G.

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Hamiltonian Properties of Iterated Line Graphs √ Define l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path in G is a path in G whose interval vertices have degree two in G. √ (Lai, 1988) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then h(G) ≤ l + 1.

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Hamiltonian Properties of Iterated Line Graphs √ Define l(G) = max{m : G has a divalent path of length m that is not both of length 2 and in a K3}, where a divalent path in G is a path in G whose interval vertices have degree two in G. √ (Lai, 1988) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then h(G) ≤ l + 1. √ (Lai etc., 2006) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then hs(G) ≤ l + 1.

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Our Results √ The s-fully cycle extendable index, fces(G), of a connected graph G is the least nonnegative integer m such that Lm(G) is s-fully cycle extendable.

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Our Results √ The s-fully cycle extendable index, fces(G), of a connected graph G is the least nonnegative integer m such that Lm(G) is s-fully cycle extendable. √ (Shao, 2011) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then fces(G) ≤    l(G) + s + 1 : if 0 ≤ s ≤ 1 l(G) + ⌊log2s⌋ + 2 : if s ≥ 2

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Sketch of the Proof

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Main Idea of the Proof √ By the definition of the graph, each vertex v ∈ V (G) generates a a clique in G corresponding to edges incident to v, and these cliques partition E(L(G).

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Main Idea of the Proof √ By the definition of the graph, each vertex v ∈ V (G) generates a a clique in G corresponding to edges incident to v, and these cliques partition E(L(G). √ Let G be a simple connected graph that is not a path, a cycle

• r K1,3, with l(G) = l. Then each of the following holds:

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Main Idea of the Proof √ By the definition of the graph, each vertex v ∈ V (G) generates a a clique in G corresponding to edges incident to v, and these cliques partition E(L(G). √ Let G be a simple connected graph that is not a path, a cycle

• r K1,3, with l(G) = l. Then each of the following holds:

√ (i) Ll(G), Ll+1(G) are triangular,moreover,Ll+s(G) is 2s−3−triangular when s ≥ 3;

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Main Idea of the Proof √ By the definition of the graph, each vertex v ∈ V (G) generates a a clique in G corresponding to edges incident to v, and these cliques partition E(L(G). √ Let G be a simple connected graph that is not a path, a cycle

• r K1,3, with l(G) = l. Then each of the following holds:

√ (i) Ll(G), Ll+1(G) are triangular,moreover,Ll+s(G) is 2s−3−triangular when s ≥ 3; √ (ii) For an integer s ≥ 2, κ(Ll+s(G)) ≥ 2s−1 + 2.

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Main Idea of the Proof √ (Shao, 2011) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then fces(G) ≤    l(G) + s + 1 : if 0 ≤ s ≤ 1 l(G) + ⌊log2s⌋ + 2 : if s ≥ 2

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Main Idea of the Proof √ (Shao, 2011) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then fces(G) ≤    l(G) + s + 1 : if 0 ≤ s ≤ 1 l(G) + ⌊log2s⌋ + 2 : if s ≥ 2 √ Case 1 s ≥ 8.

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Main Idea of the Proof √ (Shao, 2011) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then fces(G) ≤    l(G) + s + 1 : if 0 ≤ s ≤ 1 l(G) + ⌊log2s⌋ + 2 : if s ≥ 2 √ Case 1 s ≥ 8. √ The l(G) + ⌊log2s⌋ + 1-th iterated line graph is almost trianglular.

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Main Idea of the Proof √ (Shao, 2011) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then fces(G) ≤    l(G) + s + 1 : if 0 ≤ s ≤ 1 l(G) + ⌊log2s⌋ + 2 : if s ≥ 2 √ Case 1 s ≥ 8. √ The l(G) + ⌊log2s⌋ + 1-th iterated line graph is almost trianglular. √ Case 2 0 ≤ s ≤ 7.

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Main Idea of the Proof √ (Shao, 2011) Let G be a simple connected graph with l(G) = l that is not a path, a cycle or a K1,3. Then fces(G) ≤    l(G) + s + 1 : if 0 ≤ s ≤ 1 l(G) + ⌊log2s⌋ + 2 : if s ≥ 2 √ Case 1 s ≥ 8. √ The l(G) + ⌊log2s⌋ + 1-th iterated line graph is almost trianglular. √ Case 2 0 ≤ s ≤ 7. √ Prove each case by using the structure of the line graph.

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Thank You

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