A subgraph of an edge-colored graph is rainbow if all of its edges have different colors. For a graph H and a positive integer n, the anti-Ramsey number f(n, H) is the maximum number of colors in an edge-coloring of Kn with no rainbow copy of H . The rainbow number rb(n, H) is the minimum number of colors such that any edge-coloring of Kn with rb(n, H) number of colors contains a rainbow copy o...

Let G be a nontrivial connected graph with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is called a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree that connects S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such that there is a ra...

In a properly edge colored graph, a subgraph using every color at most once is called rainbow. In this thesis, we study rainbow cycles and paths in proper edge colorings of complete graphs, and we prove that in every proper edge coloring of Kn, there is a rainbow path on (3/4− o(1))n vertices, improving on the previously best bound of (2n + 1)/3 from [?]. Similarly, a k-rainbow path in a proper...

For almost 20 years, gay rights advocates and defenders of military anti-gay discrimination engaged in a phony debate about whether allowing open service would undermine unit cohesion. To be sure, a preponderance of evidence showed that open service would not undermine cohesion, and the repeal of don't ask, don't tell (DADT) required advocates to prevail on that point in the court of public opi...

In this article, a data-driven approach was adopted to demonstrate how real-life diary techniques [ie, the experience sampling method (ESM)] could be deployed for assessment purposes in patients with psychotic disorder, delivering individualized and clinically relevant information. The dataset included patients in an acute phase of psychosis and the focus was on paranoia as one of the main psyc...

The Rainbow k-Coloring problem asks whether the edges of a given graph can be colored in k colors so that every pair of vertices is connected by a rainbow path, i.e., a path with all edges of different colors. Our main result states that for any k ≥ 2, there is no algorithm for Rainbow k-Coloring running in time 2 3/2), unless ETH fails. Motivated by this negative result we consider two paramet...

Let G = (V (G), E(G)) be a nontrivial connected graph of order n with an edge-coloring c : E(G) → {1, 2, . . . , q}, q ∈ N, where adjacent edges may be colored the same. A tree T in G is a rainbow tree if no two edges of T receive the same color. For a vertex set S ⊆ V (G), a tree connecting S in G is called an S-tree. The minimum number of colors that are needed in an edge-coloring of G such t...

Recent evidence suggests that sleep problems are associated with psychotic like experiences including paranoia. However, the mechanisms underpinning this association are not well understood and thus studies modelling hypothesised mediating factors are required. Alexithymia, the inability to recognise and describe emotions within the self may be an important candidate. In two separate studies we...

Research increasingly assesses momentary changes in paranoia in order to elucidate causal mechanisms. Observed or manipulated changes in postulated causal factors should result in fluctuations in state paranoid ideation. Previous studies often employed a state-adapted Paranoia Checklist (Freeman et al., 2005) to measure state paranoia. This study examined whether the Paranoia Checklist or subse...

Abstract: A 2-rainbow domination function of a graph G = (V, E) is a function f mapping each vertex v to a subset of {1, 2} such that ⋃ u∈N(v) f (u) = {1, 2} when f (v) = �, where N(v) is the open neighborhood of v. The weight of f is denoted by wf (G) = ∑ v∈V �f (v)�. The 2-rainbow domination number, denoted by r2(G), is the smallest wf (G) among all 2-rainbow domination functions f of G. The ...