The packing chromatic number ??(G) of a graph G is the smallest integer k for which there exists vertex coloring ?:V(G)?{1,2,…,k} such that any two vertices color i are at distance least i+1. For g?{6,8,12}, (q+1,g)-Moore graphs (q+1)-regular with girth g incidence symmetric generalized g?2-gons order q. In this paper we study G. g=6 present exact value ??(G). g=8, determine in terms intersecti...

A majority coloring of a directed graph is vertex-coloring in which every vertex has the same color as at most half its out-neighbors. Kreutzer, Oum, Seymour, van der Zypen and Wood proved that digraph $4$-coloring conjectured admits $3$-coloring. They observed Local Lemma implies conjecture for digraphs large enough minimum out-degree if, crucially, maximum in-degree bounded by a(n exponential...

Background Understanding environmental factors affecting the educational process and including them in educational planning will lead into increased outcome of education, mental health, reduced stress, and higher education performance. This study was conducted to investigate the impact of educational spaces coloring on learning and academic achievement of elementary school students in the acad...

The present two-part study examined the effects of coloring mandalas on state anxiety reduction with considerations of duration spent coloring and Big-5 personality traits. It was hypothesized that coloring mandalas for 30 minutes would significantly reduce levels of state anxiety. There were three coloring conditions that were randomly assigned to 105 participants. Measures included a Spielber...

Mindfulness has been associated with the use of coloring books for adults; however, the question of whether they do increase mindfulness has not been addressed. In two studies, we attempted to identify whether mindfulness is increased, and whether there is a need for ongoing guidance while coloring, similar to mindfulness meditation. In the first randomized controlled experiment, university stu...

This lecture deals with the problem of proper vertex colorings of graphs. More speciically, we are interested in coloring a given 3-colorable graph with as few colors as we can. It is easy to see that it is NP-Hard to nd a 3 coloring for any given 3-colorable graph-this would enable deciding whether a general graph is 3-colorable. If a polynomial-time algorithm existed for 3-coloring a 3-colora...

We define a new type of vertex coloring which generalizes vertex coloring in graphs, hypergraphs, and simplicial complexes. This coloring also generalizes oriented coloring, acyclic coloring, and star coloring. There is an associated symmetric function in noncommuting variables for which we give a deletion-contraction formula. In the case of graphs this symmetric function in noncommuting variab...

Distributed Greedy Coloring is an interesting and intuitive variation of the standard Coloring problem. Given an order among the colors, a coloring is said to be greedy if there does not exist a vertex for which its associated color can be replaced by a color of lower position in the fixed order without violating the property that neighbouring vertices must receive different colors. We consider...

Given a nite directed graph, a coloring of its edges turns the graph into a nite-state automaton. A k-synchronizing word of a deterministic automaton is a word in the alphabet of colors at its edges that maps the state set of the automaton at least on k-element subset. A coloring of edges of a directed strongly connected nite graph of a uniform outdegree (constant outdegree of any vertex) is k-...