My research focuses on the flow problems consisting of two parts, vector flows in graphs and integer flows in signed graphs. The concept of integer flows was first introduced by Tutte (1949) as a refinement of map coloring. In fact, integer flows is the dual concept of map coloring for planar graphs. This is often referred as duality theorem. Tutte proposed three celebrated flow conjectures whi...

We develop four constructions for nowhere-zero 5-ows of 3-regular graphs which satisfy special structural conditions. Using these constructions we show a minimal counterexample to Tutte's 5-ow conjecture is of order 44 and therefore every bridgeless graph of nonorientable genus 5 has a nowhere-zero 5-ow. One of the structural properties is formulated in terms of the structure of the multigraph ...

A graph G is k-triangular if each edge of G is in at least k triangles. It is conjectured that every 4-edge-connected 1-triangular graph admits a nowhere-zero Z3-flow. However, it has been proved that not all such graphs are Z3-connected. In this paper, we show that every 4-edge-connected 2-triangular graph is Z3-connected. The result is best possible. This result provides evidence to support t...

An edge uv in a graph Γ is directionally 2-signed (or, (2, d)signed) by an ordered pair (a, b), a, b ∈ {+, −}, if the label l(uv) = (a, b) from u to v, and l(vu) = (b, a) from v to u. Directionally 2-signed graphs are equivalent to bidirected graphs, where each end of an edge has a sign. A bidirected graph implies a signed graph, where each edge has a sign. We extend a theorem of Sriraj and Sam...

Graph-theoretic models have come to the forefront as some of the most powerful and practical methods for sequence assembly. Simultaneously, the computational hardness of the underlying graph algorithms has remained open. Here we present two theoretical results about the complexity of these models for sequence assembly. In the first part, we show sequence assembly to be NP-hard under two differe...

We consider cell colorings of drawings graphs in the plane. Given a multi-graph $G$ together with drawing $\Gamma(G)$ plane only finitely many crossings, we define $k$-coloring to be coloring maximal connected regions drawing, cells, $k$ colors such that adjacent cells have different colors. By $4$-color theorem, every bridgeless graph has $4$-coloring. A is $2$-colorable if and underlying Eule...