On balanced claw designs of complete multi-partite graphs

AbstractIn this paper, it is shown that a necessary and sufficient condition for the existence of a balanced claw design BCD(m, n, c, λ) of a complete m-partite graph λKm(n, n,…,n) is λ(m - 1)n ≡ 0 (mod 2c) and (m - 1)n ⩾ c

Hamiltonicity, Pancyclicity, and Cycle Extendability in Graphs

The study of cycles, particularly Hamiltonian cycles, is very important in many applications. Bondy posited his famous metaconjecture, that every condition sufficient for Hamiltonicity actually guarantees a graph is pancyclic. Pancyclicity is a stronger structural property than Hamiltonicity. An even stronger structural property is for a graph to be cycle extendable. Hendry conjectured that any graph which is Hamiltonian and chordal is cycle extendable. In this dissertation, cycle extendability is investigated and generalized. It is proved that chordal 2-connected K1,3-free graphs are cycle extendable. S-cycle extendability was defined by Beasley and Brown, where S is any set of positive integers. A conjecture is presented that Hamiltonian chordal graphs are {1, 2}-cycle extendable. Dirac’s Theorem is an classic result establishing a minimum degree condition for a graph to be Hamiltonian. Ore’s condition is another early result giving a sufficient condition for Hamiltonicity. In this dissertation, generalizations of Dirac’s and Ore’s Theorems are presented. The Chvatal-Erdos condition is a result showing that if the maximum size of an independent set in a graph G is less than or equal to the minimum number of vertices whose deletion increases the number of components of G, then G is Hamiltonian. It is proved here that the Chvatal-Erdos condition guarantees that a graph is cycle extendable. It is also shown that a graph having a Hamiltonian elimination ordering is cycle extendable. The existence of Hamiltonian cycles which avoid sets of edges of a certain size and certain subgraphs is a new topic recently investigated by Harlan, et al., which clearly has applications to scheduling and communication networks among other things. The theory is extended here to bipartite graphs. Specifically, the conditions for the existence of a Hamiltonian cycle that avoids edges, or some subgraph of a certain size, are determined for the bipartite case. Briefly, this dissertation contributes to the state of the art of Hamiltonian cycles, cycle extendability and edge and graph avoiding Hamiltonian cycles, which is an important area of graph theory

The Strong Perfect Graph Conjecture: 40 years of Attempts, and its Resolution

International audienceThe Strong Perfect Graph Conjecture (SPGC) was certainly one of the most challenging conjectures in graph theory. During more than four decades, numerous attempts were made to solve it, by combinatorial methods, by linear algebraic methods, or by polyhedral methods. The first of these three approaches yielded the first (and to date only) proof of the SPGC; the other two remain promising to consider in attempting an alternative proof. This paper is an unbalanced survey of the attempts to solve the SPGC; unbalanced, because (1) we devote a signicant part of it to the 'primitive graphs and structural faults' paradigm which led to the Strong Perfect Graph Theorem (SPGT); (2) we briefly present the other "direct" attempts, that is, the ones for which results exist showing one (possible) way to the proof; (3) we ignore entirely the "indirect" approaches whose aim was to get more information about the properties and structure of perfect graphs, without a direct impact on the SPGC. Our aim in this paper is to trace the path that led to the proof of the SPGT as completely as possible. Of course, this implies large overlaps with the recent book on perfect graphs [J.L. Ramirez-Alfonsin and B.A. Reed, eds., Perfect Graphs (Wiley & Sons, 2001).], but it also implies a deeper analysis (with additional results) and another viewpoint on the topic

Combinatorics

[no abstract available

Graph Theory

Graph theory is a rapidly developing area of mathematics. Recent years have seen the development of deep theories, and the increasing importance of methods from other parts of mathematics. The workshop on Graph Theory brought together together a broad range of researchers to discuss some of the major new developments. There were three central themes, each of which has seen striking recent progress: the structure of graphs with forbidden subgraphs; graph minor theory; and applications of the entropy compression method. The workshop featured major talks on current work in these areas, as well as presentations of recent breakthroughs and connections to other areas. There was a particularly exciting selection of longer talks, including presentations on the structure of graphs with forbidden induced subgraphs, embedding simply connected 2-complexes in 3-space, and an announcement of the solution of the well-known Oberwolfach Problem

On the Stability of Distribution Topologies in Peer-to-Peer Live Streaming Systems

Peer-to-Peer Live-Streaming-Systeme sind ständigen Störungen ausgesetzt.Insbesondere ermöglichen unzuverlässige Teilnehmer Ausfälle und Angriffe, welche überraschend Peers aus dem System entfernen. Die Folgen solcher Vorfälle werden großteils von der Verteilungstopologie bestimmt, d.h. der Kommunikationsstruktur zwischen den Peers.In dieser Arbeit analysieren wir Optimierungsprobleme welche bei der Betrachtung von Stabilitätsbegriffen für solche Verteilungstopologien auftreten. Dabei werden sowohl Angriffe als auch unkoordinierte Ausfälle berücksichtigt.Zunächst untersuchen wir die Berechnungskomplexität und Approximierbarkeit des Problems resourcen-effiziente Angriffe zu bestimmen. Dies demonstriert Beschränkungen in den Planungsmöglichkeiten von Angreifern und zeigt inwieweit die Topologieparameter die Schwierigkeit solcher Angriffsrobleme beeinflussen. Anschließend studieren wir Topologieformationsprobleme. Dabei sind Topologieparameter vorgegeben und es muss eine passende Verteilungstopologie gefunden werden. Ziel ist es Topologien zu erzeugen, welche den durch Angriffe mit beliebigen Parametern erzeugbaren maximalen Schaden minimieren.Wir identifizieren notwendige und hinreichende Eigenschaften solcher Verteilungstopologien. Dies führt zu mathematisch fundierten Zielstellungen für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen.Wir zeigen zwei große Klassen effizient konstruierbarer Verteilungstopologien, welche den maximal möglichen, durch Angriffe verursachten Paketverlust minimieren. Zusätzlich beweisen wir, dass die Bestimmung dieser Eigenschaft für beliebige Topologien coNP-vollständig ist.Soll die maximale Anzahl von Peers minimiert werden, bei denen ein Angriff zu ungenügender Stream-Qualität führt, ändern sich die Anforderungen an Verteilungstopologien. Wir zeigen, dass dieses Topologieformationsproblem eng mit offenen Problemen aus Design- und Kodierungstheorie verwandt ist.Schließlich analysieren wir Verteilungstopologien die den durch unkoordinierte Ausfälle zu erwartetenden Paketverlust minimieren. Wir zeigen Eigenschaften und Existenzbedingungen. Außerdem bestimmen wir die Berechnungskomplexität des Auffindens solcher Topologien. Unsere Ergebnisse liefern Richtlinien für das Topologie-Management von Peer-to-Peer Live-Streaming-Systemen und zeigen auf, welche Stabilitätsziele effizient erreicht werden können.The stability of peer-to-peer live streaming systems is constantly challenged. Especially, the unreliability and vulnerability of their participants allows for failures and attacks suddenly disabling certain sets of peers. The consequences of such events are largely determined by the distribution topology, i.e., the pattern of communication between the peers.In this thesis, we analyze a broad range of optimization problems concerning the stability of distribution topologies. For this, we discuss notions of stability against both attacks and failures.At first, we investigate the computational complexity and approximability of finding resource-efficient attacks. This allows to point out limitations of an attacker's planning capabilities and demonstrates the influence of the chosen system parameters on the hardness of such attack problems.Then, we turn to study topology formation problems. Here, a set of topology parameters is given and the task consists in finding an eligible distribution topology. In particular, it has to minimize the maximum damage achievable by attacks with arbitrary attack parameters.We identify necessary and sufficient conditions on attack-stable distribution topologies. Thereby, we give mathematically sound guidelines for the topology management of peer-to-peer live streaming systems.We find large classes of efficiently-constructable topologies minimizing the system-wide packet loss under attacks. Additionally, we show that determining this feature for arbitrary topologies is coNP-complete.Considering topologies minimizing the maximum number of peers for which an attack leads to a heavy decrease in perceived streaming quality, the requirements change. Here, we show that the corresponding topology formation problem is closely related to long-standing open problems of Design and Coding Theory.Finally, we study topologies minimizing the expected packet loss due to uncoordinated peer failures. We investigate properties and existence conditions of such topologies. Furthermore, we determine the computational complexity of constructing them.Our results provide guidelines for the topology management of peer-to-peer live streaming systems and mathematically determine which goals can be achieved efficiently

Harnessing rare category trinity for complex data

In the era of big data, we are inundated with the sheer volume of data being collected from various domains. In contrast, it is often the rare occurrences that are crucially important to many high-impact domains with diverse data types. For example, in online transaction platforms, the percentage of fraudulent transactions might be small, but the resultant financial loss could be significant; in social networks, a novel topic is often neglected by the majority of users at the initial stage, but it could burst into an emerging trend afterward; in the Sloan Digital Sky Survey, the vast majority of sky images (e.g., known stars, comets, nebulae, etc.) are of no interest to the astronomers, while only 0.001% of the sky images lead to novel scientific discoveries; in the worldwide pandemics (e.g., SARS, MERS, COVID19, etc.), the primary cases might be limited, but the consequences could be catastrophic (e.g., mass mortality and economic recession). Therefore, studying such complex rare categories have profound significance and longstanding impact in many aspects of modern society, from preventing financial fraud to uncovering hot topics and trends, from supporting scientific research to forecasting pandemic and natural disasters. In this thesis, we propose a generic learning mechanism with trinity modules for complex rare category analysis: (M1) Rare Category Characterization - characterizing the rare patterns with a compact representation; (M2) Rare Category Explanation - interpreting the prediction results and providing relevant clues for the end-users; (M3) Rare Category Generation - producing synthetic rare category examples that resemble the real ones. The key philosophy of our mechanism lies in "all for one and one for all" - each module makes unique contributions to the whole mechanism and thus receives support from its companions. In particular, M1 serves as the de-novo step to discover rare category patterns on complex data; M2 provides a proper lens to the end-users to examine the outputs and understand the learning process; and M3 synthesizes real rare category examples for data augmentation to further improve M1 and M2. To enrich the learning mechanism, we develop principled theorems and solutions to characterize, understand, and synthesize rare categories on complex scenarios, ranging from static rare categories to time-evolving rare categories, from attributed data to graph-structured data, from homogeneous data to heterogeneous data, from low-order connectivity patterns to high-order connectivity patterns, etc. It is worthy of mentioning that we have also launched one of the first visual analytic systems for dynamic rare category analysis, which integrates our developed techniques and enables users to investigate complex rare categories in practice