A proper edge coloring of a graph G is called acyclic if there is no 2-colored cycle in G. The acyclic edge chromatic number of G is the least number of colors in an acyclic edge coloring of G. In this paper, it is proved that the acyclic edge chromatic number of a planar graph G is at most ∆(G)+2 if G contains no i-cycles, 4≤ i≤ 8, or any two 3-cycles are not incident with a common vertex and ...

A hypergraph is color-critical if deleting any edge or vertex reduces the chromatic number; a color-critical hypergraph with chromatic number k is k-critical. Every k-chromatic hypergraph contains a k-critical hypergraph, so one can study chromatic number by studying the structure of k-critical (hyper)graphs. There is vast literature on k-critical graphs and hypergraphs. Many references can be ...

The chromatic number of a graph G, denoted by χ(G), is the minimum number of colors such that G can be colored with these colors in such a way that no two adjacent vertices have the same color. A clique in a graph is a set of mutually adjacent vertices. The maximum size of a clique in a graph G is called the clique number of G. The Turán graph Tn(k) is a complete k-partite graph whose partition...

Form vision is traditionally regarded as processing primarily achromatic information. Previous investigations into the statistics of color and luminance in natural scenes have claimed that luminance and chromatic edges are not independent of each other and that any chromatic edge most likely occurs together with a luminance edge of similar strength. Here we computed the joint statistics of lumi...

A colouring of the vertices of a graph (or hypergraph) G is adapted to a given colouring of the edges of G if no edge has the same colour as both (or all) its vertices. The adaptable chromatic number of G is the smallest integer k such that each edge-colouring of G by colours 1, 2, . . . , k admits an adapted vertex-colouring of G by the same colours 1, 2, . . . , k. (The adaptable chromatic nu...

A strong edge coloring of a graph G is a proper edge coloring in which every color class is an induced matching. The strong chromatic index χs(G) of a graph G is the minimum number of colors in a strong edge coloring of G. In this note, we improve a result by Dębski et al. [Strong chromatic index of sparse graphs, arXiv:1301.1992v1] and show that the strong chromatic index of a k-degenerate gra...

We make the following three observations regarding a question popularized by Katznelson: is every subset of $${\mathbb {Z}}$$ which set Bohr recurrence also topological recurrence? (i) If G countable abelian group and $$E\subseteq G$$ an $$I_0$$ set, then $$E-E$$ recurrence. In particular $$\{2^n-2^m : n,m\in {\mathbb {N}}\}$$ (ii) Let {Z}}^{\omega }$$ be direct sum countably many copies with s...

Although many colour-depth phenomena are predictable from the interocular difference in monocular chromatic diplopia caused by the eye's transverse chromatic aberration (TCA), several reports in the literature suggest that other factors may also be involved. To test the adequacy of the optical model under a variety of conditions, we have determined experimentally the effects of background colou...

The notion of (circular) colorings of edge-weighted graphs is introduced. This notion generalizes the notion of (circular) colorings of graphs, the channel assignment problem, and several other optimization problems. For instance, its restriction to colorings of weighted complete graphs corresponds to the traveling salesman problem (metric case). It also gives rise to a new definition of the ch...

An additive hereditary property of graphs is a class of simple graphs which is closed under unions, subgraphs and isomorphisms. Let P and Q be hereditary properties of graphs. The generalized edge-chromatic number ρQ(P) is defined as the least integer n such that P ⊆ nQ. We investigate the generalized edge-chromatic numbers of the properties → H, Ik, Ok, W∗ k , Sk and Dk.