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 √ 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 . – 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 √ 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 – 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 – 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 : 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 L 0 ( G ) = G and for any integer k > 0 , L k ( G ) = L ( L k − 1 ( G )) . – p. 5/15
Iterated Line Graphs √ For a nontrivial connected graph G , we define L 0 ( G ) = G and for any integer k > 0 , L k ( G ) = L ( L k − 1 ( G )) . √ G q � � v 0 v 1 v 2 q q q ❅ ❅ q – p. 5/15
Iterated Line Graphs √ For a nontrivial connected graph G , we define L 0 ( G ) = G and for any integer k > 0 , L k ( G ) = L ( L k − 1 ( G )) . √ G q � � v 0 v 1 v 2 q q q ❅ ❅ q √ L ( G ) q � � q q ❅ ❅ q – p. 5/15
Iterated Line Graphs √ For a nontrivial connected graph G , we define L 0 ( G ) = G and for any integer k > 0 , L k ( G ) = L ( L k − 1 ( G )) . √ G q � � v 0 v 1 v 2 q q q ❅ ❅ q √ L ( G ) q � � q q ❅ ❅ q √ L 2 ( 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 L i ( G ) is [2 i − 1 ( k − 2) + 2] -connected and [2 i ( k − 2) + 2] -edge-connected. – p. 7/15
Early Results √ (Chartrand and Stewart, 1969) If G is k -connected, then L i ( G ) is [2 i − 1 ( k − 2) + 2] -connected and [2 i ( k − 2) + 2] -edge-connected. – p. 8/15
Early Results √ (Chartrand and Stewart, 1969) If G is k -connected, then L i ( G ) is [2 i − 1 ( k − 2) + 2] -connected and [2 i ( 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 , ∆( L i +1 ( G )) = 2∆( L i ( G )) − 2 . – p. 8/15
Early Results √ (Chartrand and Stewart, 1969) If G is k -connected, then L i ( G ) is [2 i − 1 ( k − 2) + 2] -connected and [2 i ( 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 , ∆( L i +1 ( G )) = 2∆( L i ( 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 ( L i +1 ( G )) = 2 δ ( L i ( G )) − 2 . – p. 8/15
Early Results √ (Chartrand and Stewart, 1969) If G is k -connected, then L i ( G ) is [2 i − 1 ( k − 2) + 2] -connected and [2 i ( 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 , ∆( L i +1 ( G )) = 2∆( L i ( 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 ( L i +1 ( G )) = 2 δ ( L i ( 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. – p. 8/15
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. – 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 √ 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 L m ( G ) is Hamiltonian. – 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. √ The hamiltonian index, h ( G ) , of a connected graph G is the least nonnegative integer m such that L m ( G ) is Hamiltonian. √ The s -hamiltonian index, h s ( G ) , of a connected graph G is the least nonnegative integer m such that L m ( 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 K 3 } , 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 K 3 } , 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 K 1 , 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 K 3 } , 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 K 1 , 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 K 1 , 3 . Then h s ( G ) ≤ l + 1 . – p. 10/15
Our Results √ The s -fully cycle extendable index, fce s ( G ) , of a connected graph G is the least nonnegative integer m such that L m ( G ) is s -fully cycle extendable. – p. 11/15
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