Jianbing Niu

Bio: Jianbing Niu is an academic researcher from Tennessee Wesleyan College. The author has contributed to research in topics: Cubic graph & Edge coloring. The author has an hindex of 1, co-authored 1 publications receiving 25 citations.

On Perfect Matching Coverings and Even Subgraph Coverings

Abstract: A perfect matching covering of a graph G is a set of perfect matchings of G such that every edge of G is contained in at least one member of it. Berge conjectured that every bridgeless cubic graph admits a perfect matching covering of order at most 5 we call such a collection of perfect matchings a Berge covering of G. A cubic graph G is called a Kotzig graph if G has a 3-edge-coloring such that each pair of colors forms a hamiltonian circuit introduced by R. Haggkvist, K. Markstrom, J Combin Theory Ser B 96 2006, 183-206. In this article, we prove that if there is a vertex w of a cubic graph G such that G-wi¾?, the graph obtained from G-w by suppressing all degree two vertices is a Kotzig graph, then G has a Berge covering. We also obtain some results concerning the so-called 5-even subgraph double cover conjecture.

On Fulkerson conjecture

Abstract: If $G$ is a bridgeless cubic graph, Fulkerson conjectured that we can find $6$ perfect matchings (a {\em Fulkerson covering}) with the property that every edge of $G$ is contained in exactly two of them. A consequence of the Fulkerson conjecture would be that every bridgeless cubic graph has $3$ perfect matchings with empty intersection (this problem is known as the Fan Raspaud Conjecture). A {\em FR-triple} is a set of $3$ such perfect matchings. We show here how to derive a Fulkerson covering from two FR-triples. Moreover, we give a simple proof that the Fulkerson conjecture holds true for some classes of well known snarks.

Berge–Fulkerson Conjecture on Certain Snarks

Abstract: A snark is a connected, bridgeless cubic graph with chromatic index equal to 4. The Berge–Fulkerson conjecture proposed in 1971 states that every bridgeless cubic graph contains a family of six perfect matchings such that each edge is contained in exactly two of them. This conjecture holds trivialy for 3-edge colorable graphs. Thus a possible minimum counterexample for the conjecture is a snark. In this paper we shown that the conjecture holds for families of snarks such as Loupekhine snarks of first and second kind and the Watkins snark. We also determine the excessive index for Loupekhine snarks of first and second kind.