ℤ 2 × ℤ 2-Cordial Cycle-Free Hypergraphs

Hovey introduced A-cordial labelings as a generalization of cordial and harmonious labelings [7]. If A is an Abelian group, then a labeling f: V(G) → A of the vertices of some graph G induces an edge labeling on G; the edge uv receives the label f(u)+f(v). A graph G is A-cordial if there is a vertex-labeling such that (1) the vertex label classes differ in size by at most one and (2) the induced edge label classes differ in size by at most one. The problem of A-cordial labelings of graphs can be naturally extended for hypergraphs. It was shown that not every 2-uniform hypertree (i.e., tree) admits a ℤ 2 × ℤ 2-cordial labeling [8]. The situation changes if we consider p-uniform hypertrees for a bigger p. We prove that a p-uniform hypertree is ℤ 2 × ℤ 2-cordial for any p > 2, and so is every path hypergraph in which all edges have size at least 3. The property is not valid universally in the class of hypergraphs of maximum degree 1, for which we provide a necessary and sufficient condition. © Sylwia Cichacz et al., published by Sciendo 2019

2-splittable and cordial graphs

E. Miller and G.E. Stevens proved in [5] the existence of certain families of 2-splittable caterpillars. In this paper we characterize other families of 2-splittable caterpillars. Moreover, we show that for some of them there exists a friendly labeling inducing two isomorphic subgraphs

Interaktywne prezentacje wybranych zagadnień z geometrii wykreślnej w programie Flash MX

The program Flash MX is designed to create, among other things, commercial Web sites and presentations that can clearly explain complex issues and research in all fields of science, art. and culture. This is a new platform for advanced interactive applications that can be distributed online and on CD-ROM. The paper describes the use of Flash MX for the presentations of selected constructions of descriptive geometry. Presented geometrie constructions have been made in the Monge, Mapping and Central projections methods. Created Computer animations are part of e-learning platform on the Internet.Program Flash MX przeznaczony jest do tworzenia m.in. komercyjnych stron WWW oraz prezentacji, które w przejrzysty sposób mogą wyjaśnić złożone zagadnienia i wyniki badań we wszystkich dziedzinach nauki, kultury i sztuki. Jest to nowa platforma dla zaawansowanych interaktywnych aplikacji, które mogą być rozpowszechniane w Internecie, jak i na płytach CD-ROM. W artykule przedstawiono zastosowanie programu Flash MX do prezentacji wybranych konstrukcji z geometrii wykreślnej. Prezentowane konstrukcje geometryczne wykonano w rzutach Monge'a, Cechowanym i Środkowym. Utworzone animacje komputerowe są elementem platformy e-learningowej dostępnej w Internecie

Open trails in digraphs

It has been shown in [S. Cichacz, A. Görlich, Decomposition of complete bipartite graphs into open trails, Preprint MD 022, (2006)] that any bipartite graph Ka,b, is decomposable into open trails of prescribed even lengths. In this article we consider the corresponding question for directed graphs. We show that the complete directed graphs ↔K n and ↔K a,b are arbitrarily decomposable into directed open trails