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Meat production Goal: Freshly prepared meat for consumers Reduce the quantity of discarded meat Solution: Predicting the quantity of meat sold per day The forecasting model for few days ahead

Background and Purpose: The survey was performed to determine the reasons that lead students to possibly commit plagiarism during their studies. By doing so, we wanted to determine the main reason for the appearance of plagiarism and how, within this main reasons, various indicators of plagiarism are judged and, finally, how demographic data and student motivation for study are associated with the reasons for plagiarism. Design/Methodology/Approach: A paper-and-pencil survey was carried out among 17 faculties of the University of Maribor in Slovenia. A sample of 139 students 85 males and 54 females participated in this study, ages ranged from 19 to 36 years. The questionnaire contained 95 closed questions referring to: (i) general data, (ii) education, (iii) social status, (iv) awareness of plagiarism, and (v) reasons for plagiarism. Parametric and nonparametric statistical tests were performed depending on distributions of the answers. Results: The results reveal that information and communication technology is largely responsible for the plagiarism with two reasons highlighted: ease of copying and ease of access to materials and new technologies. We also found some differences between low and high motivated students. Different average values of the answers considering motivation for study were confirmed for academic skills, teaching factors and other reasons for plagiarism, where the average for lower motivated students is significantly different (higher) than the average for higher motivated students. At the end we could find no direct relationship between the average time spent on the Internet and plagiarism. Conclusion: The transmission of knowledge is the basic mission of faculties. This mission is based on moral beliefs about the harmfulness of its abuse, and plagiarism is exactly such abuse. Regardless of the students past at this point professors are those who could greatly contribute to the right set of skills to keep students off plagiarising. Ozadje in namen: Namen raziskave je bil ugotoviti, kateri so tisti vzroki zaradi katerih se študenti odločijo za plagia­rizem. Želeli smo dognati, kateri je glavni vzrok, da do plagiarizma pride in kako so znotraj glavnega vzroka ocenjene posamezne trditve. Analizirali smo tudi povezanost demografskih podatkov študentov in njihovo stopnjo motivacije za študij z vzroki za plagiarizem. Oblikovanje/metodologija/pristop: Podatke za raziskavo smo zbrali z anketnim vprašalnikom, ki smo ga razdelili med študente 17 fakultet Univerze v Mariboru v Sloveniji. V raziskavi je sodelovalo 139 študentov, 85 moških in 54 žensk, starih od 19 do 36 let. Anketni vprašalnik je vseboval 95 zaprtih vprašanj, ki so bila strukturirana v: (i) splošne podatke, (ii) izobraževanje, (iii) socialni status, (iv) zavedanje plagiatorstva, (v) vzroki za plagiarizem. Odgovori so bili porazdeljeni na likertovi lestvici od (1) sploh se ne strinjam do (5) močno se strinjam. Izvedeni so bili parametrični in neparametrični statistični testi. Rezultati: Rezultati kažejo, da je informacijska in komunikacijska tehnologija v veliki meri odgovorna za plagiarizem študentov. V največji meri sta izpostavljena vzroka: enostavnost kopiranja in preprost dostop do materialov in novih tehnologij. Ugotovili smo tudi nekatere razlike med nizko in visoko motiviranimi študenti za študij. Razlike v povprečnih vrednostih odgovorov pri stopnji motiviranosti za študij se kažejo pri vzrokih: akademske spretnosti, učni dejavniki in drugi razlogi za plagiarizem, pri čemer so povprečne vrednosti nižje motiviranih študentov statistično pomembno dru­gačne (višje) od povprečnih vrednosti višje motiviranih študentov. Med povprečnim časom, ki ga študenti porabijo na internetu in vzroki za plagiarizem ni bilo ugotovljene neposredne povezanosti. Zaključek: Osnovno poslanstvo fakultet je prenos znanja na študente. To poslanstvo temelji na moralnem in etično sprejemljivem ravnanju vseh subjektov. Fakulteta je tista, ki mora študente usmerjati ter jim nuditi pravi nabor spret­nosti in veščin, da do plagiarizma ne bo prihajalo.

Diplomsko delo obravnava družine 2m-regularnih grafov in m-regularnih dvodelnih grafov, ki se dekomponirajo v izomorfne kopije drevesa T z m povezavami. Obravnavana je k-terica r1,..., rk z vsoto i=1 do k ri = m. Za drevo T s takim k-barvanjem po povezavah z r i povezavami barve i, da ima vsaka pot v T eno ali dve povezavi iste barve, je predstavljen rezultat, da se vsak kartezični produkt grafov G1,...,Gk, kjerje Gi 2ri-regularen graf za 1

Diplomsko delo obravnava povezano particijsko dimenzijo grafov. Tvorijo ga tri poglavja. V prvem poglavju so predstavljeni osnovni pojmi, definicije in primeri iz teorije grafov. Drugo poglavje je namenjeno predstavitvi povezane particijske dimenzije grafov in njenih lastnosti. Obravnavana je povezava med particijsko dimenzijo in povezano particijsko dimenzijo grafov. Podana je karakterizacija grafov reda n, katerih povezana particijska dimenzija je enaka 2, n ali n-1. V tretjem poglavju je določena povezana particijska dimenzija dreves, koles in Jahangirovih grafov. Poleg tega je dokazan izrek, ki pravi, da za vsak par celih števil a in b, kjer je a večje ali enako 3 in b manjše ali enako 2a-1 in hkrati večje od a, obstaja povezan graf G, da je pd(G)=a in cpd(G)=b. The diploma paper deals with the connected partition dimension of graphs. It consists of three chapters. The first chapter presents the basic concepts, definitions and examples from the graph theory. The second chapter presents connected partition dimension of graphs and its properties. It deals with the relationship between the partition dimension of graphs and the connected partition dimension of graphs. The characterization of graphs of order n, for which the connected partition dimension is equal to 2, n or n – 1 is given. In the third chapter, the connected partition dimension of trees, wheels and Jahangir graphs is determined. Moreover, the proof of the theorem which states that for every pair of integers a and b, with a greater than or equal to 3 and b less than or equal to 2a-1 and b greater than a, there is a connected graph G having pd(G)=a and cpd(G)=b is given.

Diplomsko delo, ki je razdeljeno na tri poglavja, predstavlja matematični model igre Enka. V prvem poglavju bomo najprej definirali in opredelili osnovne pojme, ki jih bomo potrebovali v nadaljevanju. V drugem poglavju bomo predstavili dva matematična modela Enke in njuni obliki ter definirali graf Enke. Sledila bo karakterizacija ENKA-1 grafov, nato bo predstavljen še graf šahovnice, ki je definiran podobno kot ENKA-1 graf. V zadnjem poglavju bomo predstavili zahtevnosti Enke z dvema igralcema in Enke z enim igralcem. Pokazali bomo, da sta pripadajoča problema ENKA-2 in ENKA-1 $NP$-polna problema. The purpose of this graduation thesis is to present mathematical model of the card game Uno and is divided into three sections. First section of the thesis defines basic concepts, which will be used throughout this study. Second section presents two mathematical models, their forms and define graph of Uno, followed by characterization of graph UNO-1. In addition, second section includes chessboard, which is defined similarly as graph UNO-1. Last section presents complexity of Uno with two players and Uno with one player. Moreover, the third section of this graduation thesis demonstrates that associated problems of UNO-2 and UNO-1 are $NP$- complete.

Diplomsko nalogo tvori šest poglavij. Po uvodu sledi poglavje z osnovnimi pojmi teorije grafov, ki so uporabljeni v diplomski nalogi. V drugem poglavju so predstavljene osnovne lastnosti igre s palicami ter definirani pojmi palično število, optimalno palično število, palična poteza, dvopalična poteza, odstranitvena poteza, razporeditev na grafu, dobra (multi) razporeditev na grafu, cilj poteze in izid razporeditve. V tretjem poglavju sta podana palično in optimalno palično število poti, ciklov in spojev grafov. Prikazani so tudi primeri razporeditve za nekatere poti in cikle manjšega reda. V četrtem poglavju sta podani palično in optimalno palično število kartezičnega produkta polnih grafov ter določeni spodnja in zgornja meja paličnega števila kartezičnega produkta $G square K_n$, ki temelji na dvopaličnem številu. V petem poglavju je določeno palično število hiperkock. Poleg tega sta določeni še spodnja in zgornja meja za optimalno palično število hiperkock. V šestem poglavju pa so določene spodnje in zgornje meje za palično in optimalno palično število grafov z majhnim premerom. The graduation thesis consists of six sections. After introduction the basic concepts of the Graph Theory used in the thesis are presented. In the second section the basic properties of pegging are introduced and the definitions of pegging number, optimal pegging number, pegging move, pebbling move, removal move, distribution on a graph, proper(multi) distribution on a graph, target vertex and reach of a distribution are given. In the third section the pegging and the optimal pegging number of paths, cycles and joins are given and some examples of distributions for paths and cycles of small order are presented. In the fourth section the pegging and the optimal pegging number of the Cartesian product of complete graphs are determined. Moreover, the lower and the upper bound for the pegging number of the Cartesian product $G square K_n$, which are based on the pebbling number, are established. In the fifth section the pegging number of the hypercubes and also the lower and the upper bound for the optimal pegging number of the hypercubes are determined. In the sixth section the lower and the upper bound for the pegging and the optimal pegging number of graphs of small diameter are presented.

Diplomsko delo obravnava particijsko dimenzijo grafov in je sestavljeno iz treh poglavij. V prvem poglavju bomo predstavili osnovne pojme iz teorije grafov in spoznali bomo štiri najbolj poznane produkte grafov, s poudarkom na kartezičnem produktu. Drugo poglavje bomo namenili predstavitvi dveh, za nas najbolj pomembnih dimenzij grafov. To sta metrična in particijska dimenzija grafov. Najprej bomo definirali metrično dimenzijo grafov in spoznali njene lastnosti. Nato se bomo posvetili particijski dimenziji grafov in njenim lastnostim. Pri obeh dimenzijah bomo za boljšo predstavitev podali tudi nekaj primerov. Na koncu tega poglavja pa si bomo še pogledali povezanost omenjenih dveh dimenzij. V zadnjem poglavju bomo definirali particijsko dimenzijo kartezičnega produkta grafov. Pogledali si bomo zgornjo mejo te dimenzije, nato bomo spoznali njeno povezavo z metrično dimenzijo in na koncu navedli še dva aktualna odprta problema. The diploma paper discusses the partition dimension of graphs and consists of three chapters. In the first chapter the basic terms of the graph theory as well as the four most widely known products of graphs, with the emphasis on the Cartesian product, are presented. The second chapter is devoted to the presentation of the two for us most important dimensions of graphs. These are the metric and the partition dimension of graphs. First of all the metric dimension of graphs and it characteristics are presented. Afterwards the emphasis is on the partition dimension of graphs and its features. For a better understanding of both dimensions some examples are added. At the end of this chapter the connections between the two dimensions are presented. In the last chapter the partition dimension of the Cartesian product of graphs is defined. The upper bound of this dimension and the connections with the metric dimension are also shown. In the end two current open problems are stated.

V diplomskem delu sta najprej predstavljeni dve osnovni igri s klobuki, in sicer Ebertova igra s klobuki in igra s klobuki v vrsti ter njune različice skupaj z rešitvami. Obe igri sta igri enega ali več igralcev, pri kateri morajo igralci na podlagi informacij o barvah soigralčevih klobukov, ugotoviti barvo svojega klobuka. V nadaljevanju sledi obravnava Nove igre s klobuki v vrsti, ki je kombinacija obeh prej omenjenih iger. Pri tej igri je obravnavana tudi optimalna strategija igre, katere dokaz temelji na kombinatoričnih situacijah. V zadnjem delu je kot zanimivost predstavljen še drugačen pristop k igri z uporabo teorije grafov, oziroma natančneje, z uporabo hiperkock. Two basic hat games - Ebert's Hat Game and Hats-on-a-line Game with its variations and solutions are presented in the first part of diploma thesis. Both are inteaded for one or more players where each has to guess the colour of own hat based only on the information about hat colours of co-players. Furthermore, the New Hats-on-a-line Game is the combination of both previously mentioned game. The optimal strategy of the game whose proof is grounded on combinatorial situations is dealt with that game. An interesting fact in the last part is a different approach to the game using graph theory or more precisely, using hypercubes.