Factorizations of complete graphs into caterpillars with diameter 6 and three rich vertices

Import 05/08/2014Cílem této bakalářské práce je nalézt alespoň jednu nekonečnou, dosud neznámou, třídu housenek s průměrem 6, které faktorizují kompletní graf K2n, kde n je liché. Zaměřili jsme se na housenky s přesně třemi vrcholy stupně alespoň 3. Housenky na 2n vrcholech, které zkoumáme, obsahují vrchol stupně n (to je největší stupeň vrcholu, který ještě umožňuje faktorizaci K2n) a vrchol stupně 3. Součástí práce je citace známých výsledků pro faktorizace kompletních grafů na housenky s průměrem 4 a 5, dále citace nutných podmnínek a postačujících podmínek pro faktorizaci kompletních grafů na kostry a jejich aplikace při charakterizaci vybrané nekonečné třídy housenek s průměrem 6.The goal of this bachelor thesis is to find at least one unknown infinite class of caterpillars with diameter 6 that factorizes a complete graph of order 2n, where n is odd. We focus into caterpillars with exactly three vertices of degree at least 3. The caterpillars, which we investigate, contain a vertex of degree n (it is a maximum degree of vertex that allows factorization of K2n) and vertex of degree 3. A part of thesis is a citation of known results for factorizations of complete graphs into caterpillars with diameters 4 and 5. Further included is a citation of necessary conditions and sufficient conditions for factorizations of complete graphs into spanning trees and their applications for characterization of chosen infinite class of caterpillars with diameter 6.470 - Katedra aplikované matematikyvelmi dobř

Balance, partial balance and balanced-type spectra in graph-designs

For a given graph G, the set of positive integers v for which a G-design exists is usually called the 'spectrum' for G and the determination of the spectrum is sometimes called the 'spectrum problem'. We consider the spectrum problem for G-designs satisfying additional conditions of 'balance', in the case where G is a member of one of the following infinite families of trees: caterpillars, stars, comets, lobsters and trees of diameter at most 5. We determine the existence spectrum for balanced G-designs, degree-balanced and partially degree-balanced G-designs, orbit-balanced G-designs. We also address the existence question for non-balanced G-designs, for G-designs which are either balanced or partially degree-balanced but not degree-balanced, for G-designs which are degree-balanced but not orbit-balanced

Factorizations of complete graphs into tadpoles

Import 03/11/2016V této diplomové práci zkoumáme faktorizace na kompletních grafů K_{4k+3} na pulce pro k<=1 (pulec je graf G, který vznikne ztotožněním libovolného vrcholu cyklu C_m s koncovým vrcholem cesty délky |V(G)|-m, kde 3&lt;=m&lt;=|V(G)|-1). Ukážeme, že každý pulec na 4k+3 vrcholech faktorizuje kompletní graf K_{4k+3}, je-li délka cyklu m=3,4,...,2k+2,2k+4,...,4k+2 pro k liché resp. m=3,4,...,2k+1,2k+3,...,4k+2 pro k sudé. Všimneme si, že chybí délky cyklu m=2k+3 pro k liché resp. m=2k+2 pro k sudé. Důkazy pro tyto délky ještě nejsou zcela dokončené, ale určitě se objeví v článku, který bude navazovat na tuto práci.In this master thesis we investigate factorizations of complete graphs K_{4k+3} into tadpoles for k<=1 (a tadpole is a graph G that arise if we glue one terminal vertex of path of length |V(G)|-m to an arbitary vertex of cycle C_m for 3&lt;=m&lt;=|V(G)|-1). We show that every tadpole with 4k+3 vertices factorizes a complete graph K_{4k+3} if lengths of cycles are m=3,4,...,2k+2,2k+4,..., 4k+2 for k odd resp. m=3,4,...,2k+1,2k+3,...,4k+2 for k even. We see that lengths of cylces m=2k+3 for k odd resp. m=2k+2 for k even are missing. Proofs of this lengths of cycles are not finished yet. But they will featured in the article that follows this thesis.470 - Katedra aplikované matematikyvýborn

Embedding rainbow trees with applications to graph labelling and decomposition

A subgraph of an edge-coloured graph is called rainbow if all its edges have distinct colours. The study of rainbow subgraphs goes back more than two hundred years to the work of Euler on Latin squares. Since then rainbow structures have been the focus of extensive research and have found applications in the areas of graph labelling and decomposition. An edge-colouring is locally k-bounded if each vertex is contained in at most k edges of the same colour. In this paper we prove that any such edge-colouring of the complete graph Kn contains a rainbow copy of every tree with at most (1−o(1))n/k vertices. As a locally k-bounded edge-colouring of Kn may have only (n−1)/k distinct colours, this is essentially tight. As a corollary of this result we obtain asymptotic versions of two long-standing conjectures in graph theory. Firstly, we prove an asymptotic version of Ringel's conjecture from 1963, showing that any n-edge tree packs into the complete graph K(2n+o(n)) to cover all but o(n^2) of its edges. Secondly, we show that all trees have an almost-harmonious labelling. The existence of such a labelling was conjectured by Graham and Sloane in 1980. We also discuss some additional applications

Chasing the Rainbow Connection: Hardness, Algorithms, and Bounds

We study rainbow connectivity of graphs from the algorithmic and graph-theoretic points of view. The study is divided into three parts. First, we study the complexity of deciding whether a given edge-colored graph is rainbow-connected. That is, we seek to verify whether the graph has a path on which no color repeats between each pair of its vertices. We obtain a comprehensive map of the hardness landscape of the problem. While the problem is NP-complete in general, we identify several structural properties that render the problem tractable. At the same time, we strengthen the known NP-completeness results for the problem. We pinpoint various parameters for which the problem is fixed-parameter tractable, including dichotomy results for popular width parameters, such as treewidth and pathwidth. The study extends to variants of the problem that consider vertex-colored graphs and/or rainbow shortest paths. We also consider upper and lower bounds for exact parameterized algorithms. In particular, we show that when parameterized by the number of colors k, the existence of a rainbow s-t path can be decided in O∗ (2k ) time and polynomial space. For the highly related problem of finding a path on which all the k colors appear, i.e., a colorful path, we explain the modest progress over the last twenty years. Namely, we prove that the existence of an algorithm for finding a colorful path in (2 − ε)k nO(1) time for some ε > 0 disproves the so-called Set Cover Conjecture.Second, we focus on the problem of finding a rainbow coloring. The minimum number of colors for which a graph G is rainbow-connected is known as its rainbow connection number, denoted by rc(G). Likewise, the minimum number of colors required to establish a rainbow shortest path between each pair of vertices in G is known as its strong rainbow connection number, denoted by src(G). We give new hardness results for computing rc(G) and src(G), including their vertex variants. The hardness results exclude polynomial-time algorithms for restricted graph classes and also fast exact exponential-time algorithms (under reasonable complexity assumptions). For positive results, we show that rainbow coloring is tractable for e.g., graphs of bounded treewidth. In addition, we give positive parameterized results for certain variants and relaxations of the problems in which the goal is to save k colors from a trivial upper bound, or to rainbow connect only a certain number of vertex pairs.Third, we take a more graph-theoretic view on rainbow coloring. We observe upper bounds on the rainbow connection numbers in terms of other well-known graph parameters. Furthermore, despite the interest, there have been few results on the strong rainbow connection number of a graph. We give improved bounds and determine exactly the rainbow and strong rainbow connection numbers for some subclasses of chordal graphs. Finally, we pose open problems and conjectures arising from our work

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum