On tetravalent half-arc-transitive graphs of girth 5

A subgroup of the automorphism group of a graph \G is said to be {\em half-arc-transitive} on \G if its action on \G is transitive on the vertex set of \G and on the edge set of \G but not on the arc set of \G. Tetravalent graphs of girths $3$ and $4$ admitting a half-arc-transitive group of automorphisms have previously been characterized. In this paper we study the examples of girth $5$. We show that, with two exceptions, all such graphs only have directed $5$-cycles with respect to the corresponding induced orientation of the edges. Moreover, we analyze the examples with directed $5$-cycles, study some of their graph theoretic properties and prove that the $5$-cycles of such graphs are always consistent cycles for the given half-arc-transitive group. We also provide infinite families of examples, classify the tetravalent graphs of girth $5$ admitting a half-arc-transitive group of automorphisms relative to which they are tightly-attached and classify the tetravalent half-arc-transitive weak metacirculants of girth $5$

Distance-unbalancedness of graphs

In this paper we propose and study a new structural invariant for graphs, called distance-unbalanced\-ness, as a measure of how much a graph is (un)balanced in terms of distances. Explicit formulas are presented for several classes of well-known graphs. Distance-unbalancedness of trees is also studied. A few conjectures are stated and some open problems are proposed.Comment: 14 pages, 3 figure

MAGNETNA DRŽAVA PRVAŠIĆA NA JUTARNJEM ČUVANJU

Magneti nas posvuda okružuju u životu. Tajanstveni su. Čarobni. Prilogom Magnetna država prvašića na jutarnjem čuvanju željela sam svojim prvašićima približiti rad magneta i njihovu nevidljivu moć. Kao učiteljica na jutarnjem čuvanju imala sam potpunu slobodu glede izbora aktivnosti. Bila sam svjesna da je dječja otvorenost za nova iskustva, planove i tajne života svemoguća, pa sam im kroz igru pokušala dočarati ovaj čudesni svijet. I upravo se putem igre odvijala spoznaja učenika. Putem nje su uvježbavali društvene vještine, razvijali govor i artikulaciju, tkali dijaloge, razvijali matematičke vještine, istraživali prirodu, upoznavali fizikalna svojstva i tako sami došli do različitih otkrića, spoznaja i utvrđenja. Istodobno su zaključivanjem, predviđanjem, formiranjem pretpostavki i istraživanjem razvijali sposobnosti, vještine, kao i prirodno-znanstvene postupke, te produbljivali znanje o prirodnim znanostima i povezivali sadržaje različitih predmeta

MAGNETNA DRŽAVA PRVAŠIĆA NA JUTARNJEM ČUVANJU

Magneti nas posvuda okružuju u životu. Tajanstveni su. Čarobni. Prilogom Magnetna država prvašića na jutarnjem čuvanju željela sam svojim prvašićima približiti rad magneta i njihovu nevidljivu moć. Kao učiteljica na jutarnjem čuvanju imala sam potpunu slobodu glede izbora aktivnosti. Bila sam svjesna da je dječja otvorenost za nova iskustva, planove i tajne života svemoguća, pa sam im kroz igru pokušala dočarati ovaj čudesni svijet. I upravo se putem igre odvijala spoznaja učenika. Putem nje su uvježbavali društvene vještine, razvijali govor i artikulaciju, tkali dijaloge, razvijali matematičke vještine, istraživali prirodu, upoznavali fizikalna svojstva i tako sami došli do različitih otkrića, spoznaja i utvrđenja. Istodobno su zaključivanjem, predviđanjem, formiranjem pretpostavki i istraživanjem razvijali sposobnosti, vještine, kao i prirodno-znanstvene postupke, te produbljivali znanje o prirodnim znanostima i povezivali sadržaje različitih predmeta

Explicit homomorphisms of hexagonal graphs to one vertex deleted Petersen graph

The problem of deciding whether an arbitrary graph G has a homomorphism into a given graph H has been widely studied and has turned out to be very difficult. Hell and Nešetril proved that the decision problem is NP-complete unless H is bipartite. We consider a restricted problem where G is an arbitrary triangle-free hexagonal graph and H is a Kneser graph or its induced subgraph. We give an explicit construction which proves that any triangle-free hexagonal graph has a homomorphism into one-vertex deleted Petersen graph

Strongly distance-balanced graphs and graph products

AbstractA graph G is strongly distance-balanced if for every edge uv of G and every i≥0 the number of vertices x with d(x,u)=d(x,v)−1=i equals the number of vertices y with d(y,v)=d(y,u)−1=i. It is proved that the strong product of graphs is strongly distance-balanced if and only if both factors are strongly distance-balanced. It is also proved that connected components of the direct product of two bipartite graphs are strongly distance-balanced if and only if both factors are strongly distance-balanced. Additionally, a new characterization of distance-balanced graphs and an algorithm of time complexity O(mn) for their recognition, where m is the number of edges and n the number of vertices of the graph in question, are given

Training at the Police Directorate Maribor

Namen prispevka: Namen prispevka je predstaviti organiziranost usposabljanja slovenskih policistov in podati rezultate raziskave o ustreznosti usposabljanja ter o prilagojenosti usposabljanj razmeram in spremembam v družbi. V prispevku je prikazano, kako se usposabljanja izvajajo v praksi in kakšno je mnenje zaposlenih na Policijski upravi Maribor o sistemu usposabljanja. Na koncu so podani tudi predlogi izboljšav obstoječega stanja. Metode: Empirična raziskava o mnenju policistov o sistemu usposabljanja je bila opravljena v letu 2014 na vzorcu 328 policistov Policijske uprave Maribor. Kvantitativna analiza pridobljenih podatkov je bila izvedena s pomočjo programskega paketa SPSS. Na koncu so bili podani odgovori na raziskovalna vprašanja. Ugotovitve: Rezultati raziskave so pokazali, da so policisti motivirani za usposabljanja, da si želijo več usposabljanj in da menijo, da jim usposabljanja pomagajo pri opravljanju dela. Potrjena je bila domneva, da se policisti določenih usposabljanj ne morejo udeležiti, kljub želji, kar nakazuje, da usposabljanja v policiji niso najbolj ustrezno načrtovana. Omejitve/uporabnost raziskave Raziskava je bila opravljena na dovolj velikem in reprezentativnem vzorcu policistov, da lahko rezultate posplošimo na celotno populacijo Policijske uprave Maribor. Da bi dobili pregled nad problematiko za celo Slovenijo, bi bilo smiselno raziskavo opraviti na več policijskih upravah v Sloveniji. Praktična uporabnost: Ugotovitve raziskave so uporabne pri načrtovanju usposabljanj v policiji. Na podlagi podanih ugotovitev bi bilo smiselno prilagoditi sistem usposabljanja. Več pozornosti bi bilo potrebno posvetiti tako ugotavljanju potreb po usposabljanjih kot tudi sami izvedbi usposabljanj. Izvirnost/pomembnost prispevka: Prispevek predstavlja ugotovitve raziskave o zadovoljstvu policistov s sistemom usposabljanja in njihovem mnenju o uporabnosti v praksi in s tem nadgrajuje dosedanja znanja s tega področja.Purpose: The purpose of this research paper is to present the organization of training methods in the Slovenian police force as well as to display the results of research on the adequacy and adaptation of training conditions, which consequently bring changes in society. The paper reveals the transition from training to practice and the opinion of staff at the Police Directorate Maribor on the training system. The improvements to this issue are proposed at the end of the study. Design/Methods/Approach: An empirical research on the opinion of the police training system was conducted in 2014 on a sample of 328 police officers of the Police Directorate Maribor. Quantitative analysis of the obtained data was performed using SPSS. In the end, answers to the research questions were given. Findings: The results reveal that the police officers are motivated and want more training. Furthermore, they believe that training lessons contribute to their work performance. The presumption, that certain police officers can not attend the training despite the wishes, was confirmed and that shows that the trainings in the police are not appropriated planned. Research Limitations / Implications: The survey conducted on a sufficiently large sample of police officers, indicates the results can be generalized to the entire population of the Police Directorate Maribor. To get an overview of the problem in the entire country, it would be reasonable to carry out a research in several police departments in Slovenia. Practical Implications: The research findings are useful in designing of police training policies/ schedules. Based on the results of the conducted research it would be reasonable to adjust the training system. More attention should be paid to identifying training needs as well as to the implementation of training itself. Originality/Value: This paper presents the conclusions of a survey on police officers’ satisfaction with police training systems and their opinion about its practical efficiency/ applicability in practice and thus contributes to the already existing knowledge in this field