Controlling precedence in sequential stimulus presentation with Euler tours

It is important to control for stimulus history in experiments probing responses to and similarity between sequentially presented stimuli. We present a method for stimulus order randomisation that guarantees identical precedence across stimuli. Generating sequences through sampling Euler tours allows for perfectly uniform stimulus history. This deconfounds the stimulus history from the present stimulus and maximises sensitivity to stimulus history effects including repetition suppression.

Some algorithmic results for finding compatible spanning circuits in edge-colored graphs

A compatible spanning circuit in a (not necessarily properly) edge-colored graph G is a closed trail containing all vertices of G in which any two consecutively traversed edges have distinct colors. Sufficient conditions for the existence of extremal compatible spanning circuits (i.e., compatible Hamilton cycles and Euler tours), and polynomial-time algorithms for finding compatible Euler tours have been considered in previous literature. More recently, sufficient conditions for the existence of more general compatible spanning circuits in specific edge-colored graphs have been established. In this paper, we consider the existence of (more general) compatible spanning circuits from an algorithmic perspective. We first show that determining whether an edge-colored connected graph contains a compatible spanning circuit is an NP-complete problem. Next, we describe two polynomial-time algorithms for finding compatible spanning circuits in edge-colored complete graphs. These results in some sense give partial support to a conjecture on the existence of compatible Hamilton cycles in edge-colored complete graphs due to Bollobás and Erdős from the 1970s.

Euler Tours a través del software Scratch: una secuencia de tareas de investigación

O presente artigo tem como objetivo apresentar uma sequência de tarefas que foi desenvolvida pelas pesquisadoras bem como os resultados da aplicação desta para alunos do quarto e quinto ano do Ensino Fundamental de uma escola municipal na cidade de Ourinhos, Brasil. Inspirada no problema histórico “As sete pontes de Königsberg”, a sequência compreende nove tarefas que foram desenvolvidas com o software Scratch. Ao tentarem resolver as tarefas, os alunos a fizeram de forma investigativa. A escolha por este problema caracterizou-se por ser de fácil contextualização com a realidade atual e ao ser adaptado ao contexto infantil, permitiu que os alunos percebessem uma utilização prática de conhecimentos aprendidos em sala de aula como: movimentação de objeto no espaço plano, estudo de vértices, arestas, números pares e ímpares, conhecimentos previstos como conteúdos dos Anos Iniciais do Ensino Fundamental. A metodologia adotada foi de natureza qualitativa. Os resultamos mostraram que os alunos conseguiram aproximarem-se dos quatro teoremas, alguns com desempenho melhor que outros. As tarefas permitiram que se expressassem oralmente e por meio de escrita, o que possibilitou reflexões sobre o objeto de estudo por parte dos alunos e das pesquisadoras.

The Number of Euler Tours of Random Directed Graphs

In this paper we obtain the expectation and variance of the number of Euler tours of a random Eulerian directed graph with fixed out-degree sequence. We use this to obtain the asymptotic distribution of the number of Euler tours of a random $d$-in/$d$-out graph and prove a concentration result. We are then able to show that a very simple approach for uniform sampling or approximately counting Euler tours yields algorithms running in expected polynomial time for almost every $d$-in/$d$-out graph. We make use of the BEST theorem of de Bruijn, van Aardenne-Ehrenfest, Smith and Tutte, which shows that the number of Euler tours of an Eulerian directed graph with out-degree sequence $\mathbf{d}$ is the product of the number of arborescences and the term $\frac{1}{|V|}[\prod_{v\in V}(d_v-1)!]$. Therefore most of our effort is towards estimating the moments of the number of arborescences of a random graph with fixed out-degree sequence.