Yehong Shao

Every 3-connected, essentially 11-connected line graph is Hamiltonian

Thomassen conjectured that every 4-connected line graph is Hamiltonian. A vertex cut X of G is essential if G - X has at least two non-trivial components. We prove that every 3-connected, essentially 11-connected line graph is Hamiltonian. Using Ryjaceks line graph closure, it follows that every 3-connected, essentially 11-connected claw-free graph is Hamiltonian.

Hamiltonicity in 3-connected claw-free graphs

Kuipers and Veldman conjectured that any 3-connected claw-free graph with order v and minimum degree δ ≥ (v+6)/10 is Hamiltonian for v sufficiently large. In this paper, we prove that if H is a 3-connected claw-free graph with sufficiently large order v, and if δ (H) ≥ (v + 5)/10, then either H is Hamiltonian, or δ(H) = (v+5)/10 and the Ryjaceks closure cl(H) of H is the line graph of a graph obtained from the Petersen graph P10 by adding (v - 15)/10 pendant edges at each vertex of P10.

On strongly Z 2s+1 -connected graphs

Abstract An orientation of a graph G is a mod ( 2 s + 1 ) -orientation if under this orientation, the net out-degree at every vertex is congruent to zero mod ( 2 s + 1 ) . If for any function b : V ( G ) → Z 2 s + 1 satisfying ∑ v ∈ V ( G ) b ( v ) ≡ 0 ( mod 2 s + 1 ) , G always has an orientation D such that the net out-degree at every vertex v is congruent to b ( v ) mod ( 2 s + 1 ) , then G is strongly Z 2 s + 1 -connected. In this paper, we prove that a connected graph has a mod ( 2 s + 1 ) -orientation if and only if it is a contraction of a ( 2 s + 1 ) -regular bipartite graph. We also proved that every ( 4 s − 1 ) -edge-connected series–parallel graph is strongly Z 2 s + 1 -connected, and every simple 4 p -connected chordal graph is strongly Z 2 s + 1 -connected.

HamiltonianN2-locally connected claw-free graphs

A graph G is N2-locally connected if for every vertex v in G, the edges not incident with v but having at least one end adjacent to v in G induce a connected graph. In 1990, Ryjác̆ek conjectured that every 3-connected N2-locally connected claw-free graph is hamiltonian. This conjecture is proved in this note.

Obstructions to a binary matroid being graphic

Bixby and Cunningham showed that a 3-connected binary matroid M is graphic if and only if every element belongs to at most two non-separating cocircuits. Likewise, Lemos showed that such a matroid M is graphic if and only if it has exactly r(M)+1 non-separating cocircuits. Hence the presence in M of either an element in at least three non-separating cocircuits, or of at least r(M)+2 non-separating cocircuits, implies that M is non-graphic. We provide lower bounds on the size of the set of such elements, and on the number of non-separating cocircuits, in such non-graphic binary matroids. A computationally efficient method for finding such lower bounds for specific minor-closed classes of matroids is given. Applications of this method and other results on sets of obstructions to a binary matroid being graphic are given.

New sufficient condition for Hamiltonian graphs

Abstract Let G be a graph, and δ ( G ) and α ( G ) be the minimum degree and the independence number of G , respectively. For a vertex v ∈ V ( G ) , d ( v ) and N ( v ) represent the degree of v and the neighborhood of v in G , respectively. A number of sufficient conditions for a connected simple graph G of order n to be Hamiltonian have been proved. Among them are the well known Dirac condition (1952) ( δ ( G ) ≥ n 2 ) and Ore condition (1960) (for any pair of independent vertices u and v , d ( u ) + d ( v ) ≥ n ). In 1984 Fan generalized these two conditions and proved that if G is a 2-connected graph of order n and max { d ( x ) , d ( y ) } ≥ n / 2 for each pair of nonadjacent vertices x , y with distance 2 in G , then G is Hamiltonian. In 1993, Chen proved that if G is a 2-connected graph of order n , and if max { d ( x ) , d ( y ) } ≥ n / 2 for each pair of nonadjacent vertices x , y with 1 ≤ | N ( x ) ∩ N ( y ) | ≤ α ( G ) − 1 , then G is Hamiltonian. In 1996, Chen, Egawa, Liu and Saito further showed that if G is a k -connected graph of order n , and if max { d ( v ) : v ∈ S } ≥ n / 2 for every independent set S of G with | S | = k which has two distinct vertices x , y ∈ S such that the distance between x and y is 2, then G is Hamiltonian. In this paper, we generalize all the above conditions and prove that if G is a k -connected graph of order n , and if max { d ( v ) : v ∈ S } ≥ n / 2 for every independent set S of G with | S | = k which has two distinct vertices x , y ∈ S satisfying 1 ≤ | N ( x ) ∩ N ( y ) | ≤ α ( G ) − 1 , then G is Hamiltonian.

Note: On s-hamiltonian-connected line graphs

A graph G is hamiltonian-connected if any two of its vertices are connected by a Hamilton path (a path including every vertex of G); and G is s-hamiltonian-connected if the deletion of any vertex subset with at most s vertices results in a hamiltonian-connected graph. In this paper, we prove that the line graph of a (t+4)-edge-connected graph is (t+2)-hamiltonian-connected if and only if it is (t+5)-connected, and for s>=2 every (s+5)-connected line graph is s-hamiltonian-connected.

Spanning cycles in regular matroids without small cocircuits

A cycle of a matroid is a disjoint union of circuits. A cycle C of a matroid M is spanning if the rank of C equals the rank of M. Settling an open problem of Bauer in 1985, Catlin in [P.A. Catlin, A reduction method to find spanning Eulerian subgraphs, J. Graph Theory 12 (1988) 29-44] showed that if G is a 2-connected graph on n>16 vertices, and if @d(G)>n5-1, then G has a spanning cycle. Catlin also showed that the lower bound of the minimum degree in this result is best possible. In this paper, we prove that for a connected simple regular matroid M, if for any cocircuit D, |D|>=max{r(M)-45,6}, then M has a spanning cycle.

Essential edge connectivity of line graphs

Abstract Let G be a graph and L ( G ) be its line graph. In 1969, Chartrand and Stewart proved that κ ′ ( L ( G ) ) ≥ 2 κ ′ ( G ) − 2 , where κ ′ ( G ) and κ ′ ( L ( G ) ) denote the edge connectivity of G and L ( G ) respectively. We show a similar relationship holds for the essential edge connectivity of G and L ( G ) , written κ e ′ ( G ) and κ e ′ ( L ( G ) ) , respectively. In this note, it is proved that if L ( G ) is not a complete graph and G does not have a vertex of degree two, then κ e ′ ( L ( G ) ) ≥ 2 κ e ′ ( G ) − 2 . An immediate corollary is that κ ( L 2 ( G ) ) ≥ 2 κ ( L ( G ) ) − 2 for such graphs G , where the vertex connectivity of the line graph L ( G ) and the second iterated line graph L 2 ( G ) are written as κ ( L ( G ) ) and κ ( L 2 ( G ) ) respectively.

On strongly -connected graphs

Abstract An orientation of a graph G is a mod ( 2 s + 1 ) -orientation if under this orientation, the net out-degree at every vertex is congruent to zero mod ( 2 s + 1 ) . If for any function b : V ( G ) → Z 2 s + 1 satisfying ∑ v ∈ V ( G ) b ( v ) ≡ 0 ( mod 2 s + 1 ) , G always has an orientation D such that the net out-degree at every vertex v is congruent to b ( v ) mod ( 2 s + 1 ) , then G is strongly Z 2 s + 1 -connected. In this paper, we prove that a connected graph has a mod ( 2 s + 1 ) -orientation if and only if it is a contraction of a ( 2 s + 1 ) -regular bipartite graph. We also proved that every ( 4 s − 1 ) -edge-connected series–parallel graph is strongly Z 2 s + 1 -connected, and every simple 4 p -connected chordal graph is strongly Z 2 s + 1 -connected.