Degree Conditions for Hamiltonian Properties of Claw-free Graphs

This thesis contains many new contributions to the field of hamiltonian graph theory, a very active subfield of graph theory. In particular, we have obtained new sufficient minimum degree and degree sum conditions to guarantee that the graphs satisfying these conditions, or their line graphs, admit a Hamilton cycle (or a Hamilton path), unless they have a small order or they belong to well-defined classes of exceptional graphs. Here, a Hamilton cycle corresponds to traversing the vertices and edges of the graph in such a way that all their vertices are visited exactly once, and we return to our starting vertex (similarly, a Hamilton path reflects a similar way of traversing the graph, but without the last restriction, so we might terminate at a different vertex). In Chapter 1, we presented an introduction to the topics of this thesis together with Ryjáček’s closure for claw-free graphs, Catlin’s reduction method, and the reduction of the core of a graph. In Chapter 2, we found the best possible bounds for the minimum degree condition and the minimum degree sums condition of adjacent vertices for traceability of 2-connected claw-free graph, respectively. In addition, we decreased these lower bounds with one family of well characterized exceptional graphs. In Chapter 3, we extended recent results about the conjecture of Benhocine et al. and results about the conjecture of Z.-H Chen and H.-J Lai. In Chapters 4, 5 and 6, we have successfully tried to unify and extend several existing results involving the degree and neighborhood conditions for the hamiltonicity and traceability of 2-connected claw-free graphs. Throughout this thesis, we have investigated the existence of Hamilton cycles and Hamilton paths under different types of degree and neighborhood conditions, including minimum degree conditions, minimum degree sum conditions on adjacent pairs of vertices, minimum degree sum conditions over all independent sets of t vertices of a graph, minimum cardinality conditions on the neighborhood union over all independent sets of t vertices of a graph, as well minimum cardinality conditions on the neighborhood union over all t vertex sets of a graph. Despite our new contributions, many problems and conjectures remain unsolved

Quantum query complexity of minor-closed graph properties

We study the quantum query complexity of minor-closed graph properties, which include such problems as determining whether an $n$-vertex graph is planar, is a forest, or does not contain a path of a given length. We show that most minor-closed properties---those that cannot be characterized by a finite set of forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To establish this, we prove an adversary lower bound using a detailed analysis of the structure of minor-closed properties with respect to forbidden topological minors and forbidden subgraphs. On the other hand, we show that minor-closed properties (and more generally, sparse graph properties) that can be characterized by finitely many forbidden subgraphs can be solved strictly faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the quantum walk search framework and give improved upper bounds for several subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page

Combinatorics

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Trapdoor commitment schemes and their applications

Informally, commitment schemes can be described by lockable steely boxes. In the commitment phase, the sender puts a message into the box, locks the box and hands it over to the receiver. On one hand, the receiver does not learn anything about the message. On the other hand, the sender cannot change the message in the box anymore. In the decommitment phase the sender gives the receiver the key, and the receiver then opens the box and retrieves the message. One application of such schemes are digital auctions where each participant places his secret bid into a box and submits it to the auctioneer. In this thesis we investigate trapdoor commitment schemes. Following the abstract viewpoint of lockable boxes, a trapdoor commitment is a box with a tiny secret door. If someone knows the secret door, then this person is still able to change the committed message in the box, even after the commitment phase. Such trapdoors turn out to be very useful for the design of secure cryptographic protocols involving commitment schemes. In the first part of the thesis, we formally introduce trapdoor commitments and extend the notion to identity-based trapdoors, where trapdoors can only be used in connection with certain identities. We then recall the most popular constructions of ordinary trapdoor protocols and present new solutions for identity-based trapdoors. In the second part of the thesis, we show the usefulness of trapdoors in commitment schemes. Deploying trapdoors we construct efficient non-malleable commitment schemes which basically guarantee indepency of commitments. Furthermore, applying (identity-based) trapdoor commitments we secure well-known identification protocols against a new kind of attack. And finally, by means of trapdoors, we show how to construct composable commitment schemes that can be securely executed as subprotocols within complex protocols

Twin-constrained Hamiltonian paths on threshold graphs: an approach to the minimum score separation problem

The Minimum Score Separation Problem (MSSP) is a combinatorial problem that has been introduced in JORS 55 as an open problem in the paper industry arising in conjunction with the cutting-stock problem. During the process of producing boxes, áat papers are prepared for folding by being scored with knives. The problem is to determine if and how a given production pattern of boxes can be arranged such that a certain minimum distance between the knives can be kept. While it was originally suggested to analyse the MSSP as a specific variant of a Generalized Travelling Salesman Problem, the thesis introduces the concept of twin-constrained Hamiltonian cycles and models the MSSP as the problem of finding a twin-constrained Hamiltonian path on a threshold graph (threshold graphs are a specific type of interval graphs). For a given undirected graph G(N,E) with an even node set N and edge set E, and a bijective function b on N that assigns to every node i in N a "twin node" b(i)6=i, we define a new graph G'(N,E') by adding the edges {i,b(i)} to E. The graph G is said to have a twin-constrained Hamiltonian path with respect to b if there exists a Hamiltonian path on G' in which every node has its twin node as its predecessor (or successor). We start with presenting some general Öndings for the construction of matchings, alternating paths, Hamiltonian paths and alternating cycles on threshold graphs. On this basis it is possible to develop criteria that allow for the construction of twin-constrained Hamiltonian paths on threshold graphs and lead to a heuristic that can quickly solve a large percentage of instances of the MSSP. The insights gained in this way can be generalized and lead to an (exact) polynomial time algorithm for the MSSP. Computational experiments for both the heuristic and the polynomial-time algorithm demonstrate the efficiency of our approach to the MSSP. Finally, possible extensions of the approach are presented

Mejker–Brejker igre na grafovima

The topic of this thesis are different variants of Maker–Breaker positional game, where two players Maker and Breaker alternatively take turns in claiming unclaimed edges/vertices of a given graph. We consider Walker–Breaker game, played on the edge set of the graph Kn. Walker, playing the role of Maker is restricted to claim her edges according to a walk, while Breaker can claim any unclaimed edge per move. The focus is on two standard games - the Connectivity game, where Walker has the goal to build a spanning tree on Kn, and the Hamilton Cycle game, where Walker has the goal to build a Hamilton cycle on Kn. We show that Walker with bias 2 can win both games even when playing against Breaker whose bias b is of the order of magnitude n= ln n. Next, we consider (1 : 1) WalkerMaker–WalkerBreaker game on E(Kn),where both Maker and Breaker are walkers and we are interested in seeing how fast WalkerMaker can build spanning tree and Hamilton cycle. Finally, we study Maker–Breaker total domination game played on the vertex set of a given graph. Two players, Dominator and Staller, alternately take turns in claiming unclaimed vertices of the graph. Staller is Maker and wins if she can claim an open neighbourhood of a vertex. Dominator is Breaker and wins if he manages to claim a total dominating set of a graph. For certain connected cubic graphs on n ≥ 6 vertices, we give the characterization of those graphs which are Dominator’s win and those which are Staller’s win.Tema istrazivanja ove disertacije su igre tipa Mejker– Brejker u kojima uˇcestvuju dva igraˇca, Mejker i Brejker, koji naizmjeniˇcno uzimaju slobodne grane/ˇcvorove datog grafa. Bavimo se Voker–Brejker igrama koje se igraju na skupu grana grafa Kn. Voker, u ulozi Mejkera, jeograniˇcen da uzima svoje grane kao da se ˇseta kroz graf, dok Brejker moˇze da uzme bilo koju slobodnu granu grafa. Fokus je na dvije standardne igre - igri povezanosti, gdje Voker ima za cilj da napravi pokrivaju´ce stablo grafa Kn i igri Hamiltonove konture, gdje Voker ima za cilj da napravi Hamiltonovu konturu. Brejker pobjeduje ako sprijeˇci Vokera u ostvarenju njegovog cilja. Pokaza´cemo da Voker sa biasom 2 moˇze da pobijedi u obje igre ˇcak i ako igra protiv Brejkera ˇciji je bias b reda n= ln n. Potom razmatramo (1 : 1) VokerMejker–VokerBrejker igre na Kn, gdje oba igraˇca, i Mejker i Brejker, moraju da biraju grane koje su dio ˇsetnje u njihovom grafu s ciljem odredivanja brze pobjedniˇce strategije VokerMejkera u igri povezanosti i igri Hamiltonove konture. Konaˇcno, istraˇzujemo Mejker–Brejker igre totalne dominacije koje se igraju na skupu ˇcvorova datog grafa. Dva igraˇca, Dom inator i Stoler naizmjeniˇcno uzimaju slobodne ˇcvorove datog grafa. Stoler je Mejker i pobjeduje ako uspije da uzme sve susjede nekog ˇcvora. Dominator je Brejker i pobjeduje ako ˇcvorovi koje uzme dok kraja igre formiraju skup totalne dominacije. Za odredene klase povezanih kubnih grafova reda n ≥ 6, dajemo karakterizaciju onih grafova na kojima Dominator pobjeduje i onih na kojima Stoler pobjeduje.