In this paper we provide some sufficient conditions for the existence of an odd or even cycle that passing a given vertex or an edge in 2-connected or 2-edge connected graphs. We provide some similar conditions for the existence of an odd or even circuit that passing a given vertex or an edge in 2-edge connected graphs. We show that if G is a 2-connected k-regular graph, k ≥ 3, then every edge ...

Several authors have studied the graphs for which every edge is a chord of a cycle; among 2-connected graphs, one characterization is that the deletion of one vertex never creates a cut-edge. Two new results: among 3-connected graphs with minimum degree at least 4, every two adjacent edges are chords of a common cycle if and only if deleting two vertices never creates two adjacent cut-edges; am...

We present the rst truly polynomial-time approximation scheme (PTAS) for the minimum-cost k-vertex-(or, k-edge-) connected spanning subgraph problem for complete Euclidean graphs in R d : Previously it was known for every positive constant " how to construct in a polynomial time a graph on a superset of the input points which is k-vertex connected with respect to the input points, and whose cos...

Tutte conjectured that every 4-edge-connected graph admits a nowhere-zero Z3-flow and Jaeger et al. [Group connectivity of graphs–a nonhomogeneous analogue of nowhere-zero flow properties, J. Combin. Theory Ser. B 56 (1992) 165-182] further conjectured that every 5-edge-connected graph isZ3-connected.These two conjectures are in general open and few results are known so far. Aweaker version of ...

It is shown that for any locally knotted edge of a 3-connected graph in S^3, there is a ball that contains all of the local knots of that edge and it is unique up to an isotopy setwise fixing the graph. This result is applied to the study of topological symmetry groups of graphs embedded in S^3. Response to Reviewers: See attachment. SPATIAL GRAPHS WITH LOCAL KNOTS ERICA FLAPAN, BLAKE MELLOR, A...

In 1972, Tutte posed the 3-Flow Conjecture: that all 4-edge-connected graphs have a nowhere-zero 3-flow. This was extended by Jaeger et al. to allow vertices prescribed, possibly nonzero difference (modulo 3) between inflow and outflow. They conjectured 5-edge-connected with prescription function 3-flow meeting prescription. Kochol showed replacing would suffice prove Conjecture Lovász both con...

For a non-trivial connected graph G, a set S ⊆ V (G) is called an edge geodetic set of G if every edge of G is contained in a geodesic joining some pair of vertices in S. The edge geodetic number g1(G) of G is the minimum order of its edge geodetic sets and any edge geodetic set of order g1(G) is an edge geodetic basis. A connected edge geodetic set of G is an edge geodetic set S such that the ...

An (a, d)-edge-antimagic total labeling of a graph G with p vertices and q edges is a bijection f from the set of all vertices and edges to the set of positive integers {1, 2, 3, . . . , p + q} such that all the edge-weights w(uv) = f(u) + f(v) + f(uv);uv ∈ E(G), form an arithmetic progression starting from a and having common difference d. An (a, d)-edgeantimagic total labeling is called a sup...

Let G1,G2 be two connected graphs. Denote the corona and the edge corona of G1, G2 by G1 ◦ G2 and G = G1 G2, respectively. In this paper, we compute the Laplacian spectrum of the corona G◦H of two arbitrary graphs G and H and the edge corona of a connected regular graph G1 and an arbitrary graph G2.

For the interconnection network topology, it is usually represented by a graph. When a network is used, processors and/or links faults may happen. Thus, it is meaningful to consider faulty networks. We consider k-regular graphs in this paper. We define a k-regular hamiltonian and hamiltonian connected graph G is super fault-tolerant hamiltonian if G remains hamiltonian after removing at most k ...