On S-Fully Cycle Extendable Line Graphs

On s-fully cycle extendable line graphs

Yehong Shao Ohio University Southern, Ironton, OH 45638

– p. 1/15 Deﬁnitions

– p. 2/15 fully cycle extendable

– p. 3/15 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.

– p. 3/15 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.

– p. 4/15 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

– p. 4/15 Line Graphs

t t d d td t d d t t

– p. 4/15 Line Graphs

t t d d td t d d t t

– p. 4/15 Line Graphs

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

– p. 4/15 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)).

– p. 5/15 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

vq0 vq1 v q2 @ @q

– p. 5/15 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

vq0 vq1 v q2 @ @q

√ L(G) q

q q @ @q

– p. 5/15 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

vq0 vq1 v q2 @ @q

√ L(G) q

q q @ @q

√ L2(G) q @ q @q @ @ q

– p. 5/15 Early Studies of Iterated Line graphs

– p. 6/15 Early Results

√ Let G be a graph.

– p. 7/15 Early Results

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

– p. 7/15 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.

– p. 7/15 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.

– p. 8/15 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 Šoltes,´ 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 Šoltes,´ 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.

– p. 8/15 Early Results

– p. 8/15 Hamiltonian Properties of Iterated Line Graphs

– p. 9/15 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.

– p. 9/15 Hamiltonian Properties of Iterated Line Graphs

√ The s-hamiltonian index, hs(G), of a connected graph G is the least nonnegative integer m such that Lm(G) is

s-Hamiltonian. – p. 9/15 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.

– p. 10/15 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.

– p. 10/15 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.

– p. 10/15 Our Results

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

– p. 11/15 Our Results

√ The s-fully cycle extendable index, fces(G),ofa 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

 l(G) + s +1 : if 0 ≤ s ≤ 1 fces(G) ≤  l(G) + ⌊log2s⌋ +2 : if s ≥ 2 

– p. 11/15 Sketch of the Proof

– p. 12/15 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).

– p. 13/15 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

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

– p. 13/15 Main Idea of the Proof

or 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;

– p. 13/15 Main Idea of the Proof

or 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.

– p. 13/15 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

 l(G) + s +1 : if 0 ≤ s ≤ 1 fces(G) ≤  l(G) + ⌊log2s⌋ +2 : if s ≥ 2 

– p. 14/15 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

 l(G) + s +1 : if 0 ≤ s ≤ 1 fces(G) ≤  l(G) + ⌊log2s⌋ +2 : if s ≥ 2  √ Case 1 s ≥ 8.

– p. 14/15 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

 l(G) + s +1 : if 0 ≤ s ≤ 1 fces(G) ≤  l(G) + ⌊log2s⌋ +2 : if s ≥ 2  √ Case 1 s ≥ 8.

√ The l(G) + ⌊log2s⌋ + 1-th iterated line graph is almost trianglular.

– p. 14/15 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

 l(G) + s +1 : if 0 ≤ s ≤ 1 fces(G) ≤  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.

– p. 14/15 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

 l(G) + s +1 : if 0 ≤ s ≤ 1 fces(G) ≤  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.

– p. 14/15 Thank You

– p. 15/15