Hehui Wu

A Proof of a Conjecture of Ohba

We prove a conjecture of Ohba that says that every graph G on at most 2i¾?G+1 vertices satisfies i¾?i¾?G=i¾?G.

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.

Decomposition of Sparse Graphs into Forests and a Graph with Bounded Degree

For a loopless multigraph G, the fractional arboricity Arb(G) is the maximum of over all subgraphs H with at least two vertices. Generalizing the Nash-Williams Arboricity Theorem, the Nine Dragon Tree Conjecture asserts that if , then G decomposes into forests with one having maximum degree at most d. The conjecture was previously proved for ; we prove it for and when and . For , we can further restrict one forest to have at most two edges in each component. For general , we prove weaker conclusions. If , then implies that G decomposes into k forests plus a multigraph (not necessarily a forest) with maximum degree at most d. If , then implies that G decomposes into forests, one having maximum degree at most d. Our results generalize earlier results about decomposition of sparse planar graphs.

Optimal strong parity edge-coloring of complete graphs

AbstractA parity walk in an edge-coloring of a graph is a walk along which each color is used an even number of times. Let p(G) be the least number of colors in an edge-coloring of G having no parity path (a parity edge-coloring). Let

Complete minors, independent sets, and chordal graphs

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On mod (2p +1)-orientations of graphs

(G) be the least number of colors in an edge-coloring of G having no open parity walk (a strong parity edge-coloring). Always

Packing of Steiner trees and S-connectors in graphs

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