The study of circular chromatic number Xc (G) of a graph G, which is a refinement of its
chromatic number, has been very active in the past decade. Many nice results are obtained …

After a brief historical account, a few simple structural theorems about plane graphs useful
for coloring are stated, and two simple applications of discharging are given. Afterwards, the …

The main theorem of this paper provides partial results on some major open problems in
graph theory, such as Tutteʼs 3-flow conjecture (from the 1970s) that every 4-edge …

A signed graph is a graph G together with an assignment of signs+ and− to all the edges of
G where Σ is the set of negative edges. Furthermore and are considered to be equivalent if …

An oriented graph is a loopless digraph with no opposite arcs. An oriented k-colouring of an
oriented graph G ⃗ is a partition of its set of vertices into k parts in such a way that no two …

We provide a “how-to” guide to the use and application of the Discharging Method. Our aim
is not to exhaustively survey results proved by this technique, but rather to demystify the …

It was conjectured by Jaeger that every 4 p-edge-connected graph admits a modulo (2 p+ 1)-
orientation (and, therefore, admits a nowhere-zero circular (2+ 1 p)-flow). This conjecture …

A graph G is (k, 0)‐colorable if its vertices can be partitioned into subsets V1 and V2 such
that in G [V1] every vertex has degree at most k, while G [V2] is edgeless. For every integer …

In this article, we introduce and study the properties of some target graphs for 2‐edge‐
colored homomorphism. Using these properties, we obtain in particular that the 2‐edge …

A circular r-coloring of a signed graph (G, σ) is to assign points of a circle of circumference r,
r≥ 2, to the vertices of G such that vertices connected by a positive edge are at circular …