Every planar map is four colorable

2 01 0 Every planar graph without adjacent short cycles is 3 - colorable

Two cycles are adjacent if they have an edge in common. Suppose that G is a planar graph, for any two adjacent cycles C1 and C2, we have |C1| + |C2| ≥ 11, in particular, when |C1| = 5, |C2| ≥ 7. We show that the graph G is 3-colorable.

Planar graphs with square or cube root are four-colorable

Every 4-regular graph is acyclically edge-6-colorable

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiamčik (1978) and later Alon, Sudakov and Zaks (2001) conjectured that a(G) ≤ ∆ + 2 for any simple graph G with maximum degree ∆. Basavaraju and Chandran (2009) show...

Every 8-uniform 8-regular hypergraph is 2-colorable

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.

Acyclically 3-Colorable Planar Graphs

In this paper we study the planar graphs that admit an acyclic 3-coloring. We show that testing acyclic 3-colorability is NP-hard, even for planar graphs of maximum degree 4, and we show that there exist infinite classes of cubic planar graphs that are not acyclically 3-colorable. Further, we show that every planar graph has a subdivision with one vertex per edge that admits an acyclic 3-colori...