Sopena in his survey [E. Sopena, The oriented chromatic number of graphs: A short survey, preprint 2013] writes, without any proof, that an oriented cycle ~ C can be colored with three colors if and only if λ( ~ C) = 0, where λ( ~ C) is the number of forward arcs minus the number of backward arcs in ~ C. This is not true. In this paper we show that ~ C can be colored with three colors if and on...

In this article we introduce (combinatorial) multi{color discrepancy and generalize some classical results from 2{color discrepancy theory to c colors. We give a recursive method that constructs c{colorings from approximations to the 2{color discrepancy. This method works for a large class of theorems like the six{standard{deviation theorem of Spencer, the Beck{Fiala theorem and the results of ...

Consider a finite set of identical computational entities that can move freely in the Euclidean plane operating in Look-Compute-Move cycles. Let p(t) denote the location of entity p at time t; entity p can see entity q at time t if at that time no other entity lies on the line segment p(t)q(t). We consider the basic problem called Mutual Visibility: starting from arbitrary distinct locations, w...

We present V , B − V, and DDO C(41 − 42) and C(42 − 45) photometry for a sample of 75 red giants down to MV = +2 in the relatively metal-rich Galactic globular cluster M71. The C(41− 42) colors reveal a bimodal distribution of CN band strengths generally anticorrelated with CH band strength as measured by the C(42 − 45) color. Both DDO colors agree well with those found in 47 Tucanae – a nearby...

A 3-consecutive C-coloring of a graph G = (V, E) is a mapping φ : V → N such that every path on three vertices has at most two colors. We prove general estimates on the maximum number χ̄3CC(G) of colors in a 3-consecutive C-coloring of G, and characterize the structure of connected graphs with χ̄3CC(G) ≥ k for k = 3 and k = 4.

An edge-colored graph is said to be swell-colored if each triangle contains exactly 1 or 3 colors but never 2 colors and if the graph contains more than one color. It is shown that a swell-colored complete graph with n vertices contains at least √ n + 1 colors. The complete graph with n 2 vertices has a swell coloring using n + 1 colors if and only if there exists a finite affine plane of order...

We prove that coloring a 3-uniform 2-colorable hypergraph with c colors is NP-hard for any constant c. The best known algorithm [20] colors such a graph using O(n1/5) colors. Our result immediately implies that for any constants k ≥ 3 and c2 > c1 > 1, coloring a k-uniform c1-colorable hypergraph with c2 colors is NP-hard; the case k = 2, however, remains wide open. This is the first hardness re...