We provide a certifying algorithm for the problem of deciding whether a P5-free graph is 3-colorable by showing there are exactly six finite graphs that are P5-free and not 3-colorable and minimal with respect to this property.

As is well known, Lovfisz Local Lemma implies that every d-uniform d-regular hyper-graph is 2-colorable, provided d > 9. We present a different proof of a slightly stronger result; every d-uniform d-regular hypergraph is 2-colorable, provided d > 8.

We show that an Eulerian triangulation of the Klein bottle has chromatic number equal to six if and only if it contains a complete graph of order six, and it is 5-colorable, otherwise. As a consequence of our proof, we derive that every Eulerian triangulation of the Klein bottle with facewidth at least four is 5-colorable.

We give a new proof showing that it is NP-hard to color a 3-colorable graph using just four colors. This result is already known [18], but our proof is novel as it does not rely on the PCP theorem, while the one in [18] does. This highlights a qualitative difference between the known hardness result for coloring 3-colorable graphs and the factor n hardness for approximating the chromatic number...

An orthogonal double cover (ODC) of the complete graph Kn by a graph G is a collection G of n spanning subgraphs of Kn, all isomorphic toG, such that any two members ofG share exactly one edge and every edge ofKn is contained in exactly two members of G. In the 1980s Hering posed the problem to decide the existence of an ODC for the case that G is an almost-Hamiltonian cycle, i.e. a cycle of le...

We show that every plane graph with maximum face size four whose all faces of size four are vertex-disjoint is cyclically 5-colorable. This answers a question of Albertson whether graphs drawn in the plane with all crossings independent are 5-colorable.

We show that any 2−factor of a cubic graph can be extended to a maximum 3−edge-colorable subgraph. We also show that the sum of sizes of maximum 2− and 3−edge-colorable subgraphs of a cubic graph is at least twice of its number of vertices.

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...

A graph is (k, d)-colorable if one can color the vertices with k colors such that no vertex is adjacent to more than d vertices of its same color. In this paper we investigate the existence of such colorings in surfaces and the complexity of coloring problems. It is shown that a toroidal graph is (3, 2)and (5, 1)-colorable, and that a graph of genus γ is (χγ/(d + 1) + 4, d)-colorable, where χγ ...