Ramsey's theorem and self-complementary graphs

AbstractIt is proved that, given any positive integer k, there exists a self-complementary graph with more than 4·214k vertices which contains no complete subgraph with k+1 vertices. An application of this result to coding theory is mentioned

Enumeration of small triangle free Ramsey Graphs

In 1930, a paper by Frank Plumpton Ramsey entitled On a Problem of Formal Logic appeared in the Proceedings of the London Mathematical Society. Although the impetus of this paper was one of mathematical logic, a far reaching combinatorial result was needed by Ramsey to achieve his objective. This combinatorial result became known as Ram \sey\u27s Theorem. One of the combinatorial structures which was developed during the study of Ramsey\u27s Theorem is that of a Ramsey graph. A Ramsey graph, denoted (k,l,n,e), is defined as an undirected graph that contains no cliques of size k, no independent sets of size I, with order n, and size e. Knowledge of Ramsey graphs is useful in the improvement of bounds and sometimes the calculation of exact values for various Ramsey number parameter situations. Straightforward enumeration of (k, I, n, e) Ramsey graphs for larger values of n is intractable with the current computing technology available. In order to produce such graphs, specialized algorithms need to be implemented. This thesis provides the theoretical background developed by Graver and Yackel [GRA68a], expanded upon by Grinstead and Roberts [GRl82a], and generalized by Radziszowski and Kreher [RAD88a, RAD88b] for the implementation of algorithms utilized for the enumeration of various Ram sey graphs. An object oriented graph manipulation package, including the above mentioned Ramsey graph enumeration algorithms, is implemented and documented. This package is utilized for the enumeration of all (3,3), (3,4), (3,5) and (3, 6) graphs. Some (3, 7) and (3, 8) also are calculated. These results duplicate and verify Ramsey graphs previously enumerated during other investigations. [RAD88a, RAD88b] In addition to these results, some newly enumerated (3,8) critical graphs, as well as some newly enumerated (3,9) graphs, including a minimum (3, 9, 26, 52) -graph are presented

Simultaneous solutions to diagonal equations over the p-adic numbers and finite fields, and some connections with combinatorics.

SIGLEAvailable from British Library Document Supply Centre-DSC:DXN020469 / BLDSC - British Library Document Supply CentreGBUnited Kingdo

Graph Partitioning With Input Restrictions

In this thesis we study the computational complexity of a number of graph partitioning problems under a variety of input restrictions. Predominantly, we research problems related to Colouring in the case where the input is limited to hereditary graph classes, graphs of bounded diameter or some combination of the two. In Chapter 2 we demonstrate the dramatic eect that restricting our input to hereditary graph classes can have on the complexity of a decision problem. To do this, we show extreme jumps in the complexity of three problems related to graph colouring between the class of all graphs and every other hereditary graph class. We then consider the problems Colouring and k-Colouring for Hfree graphs of bounded diameter in Chapter 3. A graph class is said to be H-free for some graph H if it contains no induced subgraph isomorphic to H. Similarly, G is said to be H-free for some set of graphs H, if it does not contain any graph in H as an induced subgraph. Here, the set H consists usually of a single cycle or tree but may also contain a number of cycles, for example we give results for graphs of bounded diameter and girth. Chapter 4 is dedicated to three variants of the Colouring problem, Acyclic Colouring, Star Colouring, and Injective Colouring. We give complete or almost complete dichotomies for each of these decision problems restricted to H-free graphs. In Chapter 5 we study these problems, along with three further variants of 3-Colouring, Independent Odd Cycle Transversal, Independent Feedback Vertex Set and Near-Bipartiteness, for H-free graphs of bounded diameter. Finally, Chapter 6 deals with a dierent variety of problems. We study the problems Disjoint Paths and Disjoint Connected Subgraphs for H-free graphs

Structural solutions to maximum independent set and related problems

In this thesis, we study some fundamental problems in algorithmic graph theory. Most natural problems in this area are hard from a computational point of view. However, many applications demand that we do solve such problems, even if they are intractable. There are a number of methods in which we can try to do this: 1) We may use an approximation algorithm if we do not necessarily require the best possible solution to a problem. 2) Heuristics can be applied and work well enough to be useful for many applications. 3) We can construct randomised algorithms for which the probability of failure is very small. 4) We may parameterize the problem in some way which limits its complexity. In other cases, we may also have some information about the structure of the instances of the problem we are trying to solve. If we are lucky, we may and that we can exploit this extra structure to find efficient ways to solve our problem. The question which arises is - How far must we restrict the structure of our graph to be able to solve our problem efficiently? In this thesis we study a number of problems, such as Maximum Indepen- dent Set, Maximum Induced Matching, Stable-II, Efficient Edge Domina- tion, Vertex Colouring and Dynamic Edge-Choosability. We try to solve problems on various hereditary classes of graphs and analyse the complexity of the resulting problem, both from a classical and parameterized point of view

Symmetry and complexity in propositional reasoning

We establish computational complexity results for a number of simple problem formulations connecting group action and prepositional formulas. The results are discussed in the context of complexity results arising from established work in the area of automated reasoning techniques which exploit symmetry