On vertex-parity edge-colorings

A vertex signature \(\pi \) of a finite graph G is any mapping \(\pi \,{:}\,V(G)\rightarrow \{0,1\}\). An edge-coloring of G is said to be vertex-parity for the pair \((G,\pi )\) if for every vertex v each color used on the edges incident to v appears in parity accordance with \(\pi \), i.e. an even or odd number of times depending on whether \(\pi (v)\) equals 0 or 1, respectively. The minimum number of colors for which \((G,\pi )\) admits such an edge-coloring is denoted by \(\chi '_p(G,\pi )\). We characterize the existence and prove that \(\chi '_p(G,\pi )\) is at most 6. Furthermore, we give a structural characterization of the pairs \((G,\pi )\) for which \(\chi '_p(G,\pi )=5\) and \(\chi '_p(G,\pi )=6\). In the last part of the paper, we consider a weaker version of the coloring, where it suffices that at every vertex, at least one color appears in parity accordance with \(\pi \). We show that the corresponding chromatic index is at most 3 and give a complete characterization for it.     

Broadcasting multiple messages in the 1-in port model in optimal time

In the 1-in port model, every vertex of a synchronous network can receive at most one message in each time unit. We consider simultaneous broadcasting of multiple messages from the same source or from distinct sources in such networks with an additional restriction that every received message can be sent out to neighbors only in the next time unit and never to already informed vertex. We use a general concept of level-disjoint partitions developed for this scenario. Here we introduce a subgraph extension technique for efficient spreading information within this concept. Surprisingly, this approach with so called biwheels leads to simultaneous broadcasting of optimal number of messages on a wide class of graphs in optimal time. In particular, we provide tight results for bipartite tori, meshes, hypercubes, Knödel graphs, circulant graphs. We also propose several open problems and conjectures.     

Quality of Life and Leisure Activities: How do Leisure Activities Contribute to Subjective Well-Being?

The quality of life is determined with objective factors and also with subjective perception of factors which influence human life. Leisure activities play a very important role in subjective well-being because they provide opportunities to meet life values and needs. Through participation in leisure activities people build social relationships, feel positive emotions, acquire additional skills and knowledge, and therefore improve their quality of life. In this report we will explore how leisure activities improve subjective well-being. We will try to distinguish among different types of leisure activities and find those which contribute more to the subjective well-being. Particularly, we will explore which leisure activities contribute to the subjective well-being of women and men of different age. Our study is based on data from a representative sample of Croatian citizens (N = 4,000), who estimated their subjective well-being and participation in various leisure activities. First, we will describe the subjective well-being of various groups of people who differ by gender and age. Afterward, we will identify important leisure activities which determine subjective well-being across age and gender groups. Overall, our results show that engagement in leisure activities contributes to subjective well being, while the pattern of important leisure activities somewhat varies across different age and gender groups.     

Tests of additivity in mixed and fixed effect two-way ANOVA models with single sub-class numbers

In variety testing as well as in psychological assessment, the situation occurs that in a two-way ANOVA-type model with only one replication per cell, analysis is done under the assumption of no interaction between the two factors. Tests for this situation are known only for fixed factors and normally distributed outcomes. In the following we will present five additivity tests and apply them to fixed and mixed models and to quantitative as well as to Bernoulli distributed data. We consider their performance via simulation studies with respect to the type-I-risk and power. Furthermore, two new approaches will be presented, one being a modification of Tukey’s test and the other being a new experimental design to test for interactions.