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 ...

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.

In this paper, we address the minimum 2-edge connected spanning subgraph problem and the graph-TSP in regular bipartite graphs. For these problems, we present new approximation algorithms, each of which finds a restricted 2-factor close to a Hamilton cycle in the first step. We first prove that every regular bipartite graph of degree at least three has a square-free 2-factor. This immediately l...

We construct highly edge-connected $r$-regular graph which do not contain $r-2$ pairwise disjoint perfect matchings. The results partially answer a question stated by Thomassen [Factorizing regular graphs, J. Comb. Theory Ser. B (2019), https://doi.org/10.1016/j.jctb.2019.05.002 (article in press)].

we find recursive formulae for the number of perfect matchings in a graph g by splitting g into subgraphs h and q. we use these formulas to count perfect matching of p hypercube qn. we also apply our formulas to prove that the number of perfect matching in an edge-transitive graph is , where denotes the number of perfect matchings in g, is the graph constructed from by deleting edges with an en...

Ryjáček introduced a closure concept in claw-free graphs based on local completion at a locally connected vertex. He showed that the closure of a graph is the line graph of a triangle-free graph. Brousek and Holub gave an analogous closure concept of claw-free graphs, called the edge-closure, based on local completion at a locally connected edge. In this paper, it is shown that the edge-closure...