We consider two problems, namely Min Split-coloring and Min Cocoloring, that generalize the classical Min Coloring problem by using not only stable sets but also cliques to cover all the vertices of a given graph. We prove the NP-hardness of some cases. We derive approximation results for Min Split-coloring and Min Cocoloring in line graphs, comparability graphs and general graphs. This provide...

A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an number times on its neighborhood. This notion was recently introduced by Petruševski and Škrekovski, who proved planar admits 9-coloring; they also conjectured 5-coloring. Shortly after, this conjecture confirmed for graphs girth at least seven Cranston; outerplanar Caro, Petruševski, Škrekovski. Bui...

Graph coloring has numerous applications and is a wellknown NP-complete problem. The goal of this paper is to survey recent results of the authors on coloring and improper coloring of sparse graphs and to point out some polynomial-time algorithms for coloring (not necessarily optimal) of graphs with bounded maximum average degree. Mathematics Subject Classification: 05C15, 05C35

An adjacent vertex distinguishing edge-coloring or an avd-coloring of a simple graph G is a proper edge-coloring of G such that no pair of adjacent vertices meets the same set of colors. We prove that every graph with maximum degree ∆ and with no isolated edges has an avd-coloring with at most ∆ + 300 colors, provided that ∆ > 1020. AMS Subject Classification: 05C15

We prove that the intersection hypergraph of a family of n pseudo-disks with respect to another family of pseudo-disks admits a proper coloring with 4 colors and a conflict-free coloring with O(log n) colors. Along the way we prove that the respective Delaunay-graph is planar. We also prove that the intersection hypergraph of a family of n regions with linear union complexity with respect to a ...

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...