Iskanje po katalogu digitalne knjižnice

Opis: We propose a new coloring game on a graph, called the independence coloring game, which is played by two players with opposite goals. The result of the game is a proper coloring of vertices of a graph G, and Alice’s goal is that as few colors as possible are used during the game, while Bob wants to maximize the number of colors. The game consists of rounds, and in round i, where i = 1, 2,, … , the players are taking turns in selecting a previously unselected vertex of G and giving it color i (hence, in each round the selected vertices form an independent set). The game ends when all vertices of G are selected (and thus colored), and the total number of rounds during the game when both players are playing optimally with respect to their goals, is called the independence game chromatic number, χig(G), of G. In fact, four different versions of the independence game chromatic number are considered, which depend on who starts a game and who starts next rounds. We prove that the new invariants lie between the chromatic number of a graph and the maximum degree plus 1, and characterize the graphs in which each of the four versions of the game invariant equals 2. We compare the versions of the independence game chromatic number among themselves and with the classical game chromatic number. In addition, we prove that the independence game chromatic number of a tree can be arbitrarily large.
Ključne besede: graph, coloring, coloring game, competition-independence game, game chromatic number, tree
Objavljeno v DKUM: 09.08.2024; Ogledov: 99; Prenosov: 10
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Opis: The masters thesis, which follows the paper Graph drawing applications in combinatorial theory of maturity models, in preparation, coauthored by the author of the thesis, introduces the tiled graphs as models of learning and maturing processes. In the thesis, we show how tiled graphs can combine graphs of learning spaces or antimatroids (partial cubes) and maturity models (total orders) to yield models of learning processes. We visualise processes with optimal drawings. In the thesis, we show NP-hardness of visualisation problems resulting from most detailed models. Further, we introduce a simpler model, which ignores the details of learning and for which the visualisation problem can be solved in a polynomial time. For the rest of the thesis, we consider this model. We describe an algorithm, which finds a drawing of an ordinal panel data graph with a minimal number of edge crossings. For this problem we further define an extremal crossing number for a chosen family of ordinal panel data. Further, we explore a certain type of random instances of ordinal panel data and the expected value of a crossing number for this type of random instances. After that, we define a problem of finding the most suitable ordering on categories in panel data, in other words finding the best maturity model to fit the data. We prove the NP-hardness of the problem and formulate an integer linear program. Master thesis consists of nine chapters. The first chapter contains known results and definitions from set and graph theory and a section of computational complexity theory (NP-hardness), which will be used throughout the thesis. The following chapters present the new theory introduced in the aforementioned paper in preparation and the needed additional results and definitions. In the last chapter we present the thesis and some selected parts of the thesis with the help of learning space theory. The chapter serves as both the overview of the thesis and the use case for the theory of learning spaces, presented in the thesis.
Ključne besede: Maturity models, learning spaces, crossing number, crossing minimisation, tile crossing number.
Objavljeno v DKUM: 14.03.2024; Ogledov: 463; Prenosov: 22
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Opis: In 1930, Kuratowski showed that �3,3 and �5 are the only two minor-minimal nonplanar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. Širáň and Kochol showed that there are infinitely many k-crossing-critical graphs for any �≥2, even if restricted to simple 3-connected graphs. Recently, 2-crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodroža-Pantić, Kwong, Doroslovački, and Pantić.
Ključne besede: crossing number, crossing-critical graph, Hamiltonian cycle
Objavljeno v DKUM: 21.12.2023; Ogledov: 461; Prenosov: 30
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Opis: Let ▫$G$▫ be a graph. We introduce the acyclic b-chromatic number of ▫$G$▫ as an analog to the b-chromatic number of ▫$G$▫. An acyclic coloring of a graph ▫$G$▫ is a map ▫$c:V(G)\rightarrow \{1,\dots,k\}$▫ such that ▫$c(u)\neq c(v)$▫ for any ▫$uv\in E(G)$▫ and the induced subgraph on vertices of any two colors ▫$i,j\in \{1,\dots,k\}$▫ induce a forest. On a set of all acyclic colorings of a graph ▫$G$▫ we define a relation whose transitive closure is a strict partial order. The minimum cardinality of its minimal element is then the acyclic chromatic number ▫$A(G)$▫ of ▫$G$▫ and the maximum cardinality of its minimal element is the acyclic b-chromatic number ▫$A_b(G)$▫ of ▫$G$▫. We present several properties of ▫$A_b(G)$▫. In particular, we derive ▫$A_b(G)$▫ for several known graph families, derive some bounds for ▫$A_b(G)$▫, compare ▫$A_b(G)$▫ with some other parameters and generalize some influential tools from b-colorings to acyclic b-colorings.
Ključne besede: acyclic b-chromatic number, acyclic coloring, b-coloring
Objavljeno v DKUM: 02.08.2023; Ogledov: 421; Prenosov: 15
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Opis: Let γ(D) denote the domination number of a digraph D and let C$_m$□C$_n$ denote the Cartesian product of C$_m$ and C$_n$, the directed cycles of length n ≥ m ≥ 3. Liu et al. obtained the exact values of γ(C$_m$□C$_n$) for m up to 6 [Domination number of Cartesian products of directed cycles, Inform. Process. Lett. 111 (2010) 36–39]. Shao et al. determined the exact values of γ(C$_m$□C$_n$) for m = 6, 7 [On the domination number of Cartesian product of two directed cycles, Journal of Applied Mathematics, Volume 2013, Article ID 619695]. Mollard obtained the exact values of γ(C$_m$□C$_n$) for m = 3k + 2 [M. Mollard, On domination of Cartesian product of directed cycles: Results for certain equivalence classes of lengths, Discuss. Math. Graph Theory 33(2) (2013) 387–394.]. In this paper, we extend the current known results on C$_m$□C$_n$ with m up to 21. Moreover, the exact values of γ(C$_n$□C$_n$) with n up to 31 are determined.
Ključne besede: domination number, Cartesian product, directed cycle
Objavljeno v DKUM: 02.09.2022; Ogledov: 611; Prenosov: 12
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