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

Edge coloring of a graph

Thesis (Master)--Izmir Institute of Technology, Mathematics, Izmir, 2004Includes bibliographical references (leaves: 35-36)Text in English; Abstract: Turkish and Englishviii, 36 leavesThe edge coloring problem is one of the fundamental problem on graphs which often appears in various scheduling problems like the le transfer problem on computer networks. In this thesis, we survey old and new results on the classical edge coloring as well as the generalized edge coloring problems. In addition, we developed some algorithms and modules by using Combinatorica package to color the edges of graphs with webMathematica which is the new web-based technology

Factorization of Graphs

PhD ThesisFor d 1; s 0; a (d; d + s)-graph is a graph whose degrees all lie in the interval fd; d + 1; :::; d + sg. For r 1; a 0; an (r; r+a)-factor of a graph G is a spanning (r; r+a)-subgraph of G. An (r; r+a)-factorization of a graph G is a decomposition of G into edge-disjoint (r; r + a)-factors. A graph is (r; r + a)-factorable if it has an (r; r + a)-factorization. For t 1, let (r; s; a; t) be the least integer such that, if d (r; s; a; t), then every (d; d + s)-simple graph G has an (r; r + a)-factorization into x (r; r + a)-factors for at least t di erent values of x. Then we show that, for r 3 odd and a 2 even, (r; s; a; t) = ( r tr+s+1 a + (t 1)r + 1 if t 2, or t = 1 and a < r + s + 1; r if t = 1 and a r + s + 1; Similarily, we have evaluated (r; s; a; t) for all other values of r; s; a and t. We call (r; s; a; t) the simple graph threshold number. A pseudograph is a graph where multiple edges and multiple loops are allowed. A loop counts two towards the degree of the vertex it is on. A multigraph here has no loops. For t 1, let (r; s; a; t) be the least integer such that, if d (r; s; a; t), then every (d; d+s)-pseudograph G has an (r; r+a)-factorization into x (r; r+a)-factors for at least t di erent values of x. We call (r; s; a; t) as the pseudograph threshold number. We have also evaluated (r; s; a; t) for all values of r, s, a and t. Note that for r 3 (r; 0; 1; 1) = 1 meaning that (r; 0; 1; 1) cannot be given a nite value. This study provides various generalisations of Petersen's theorem that \Every 2k-regular graph is 2- factorable".

Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing

A graph $G$ is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from $G$ to any graph that is isomorphic to $G$. We say that $G=(V,E)$ is robustly self-ordered if the size of the symmetric difference between $E$ and the edge-set of the graph obtained by permuting $V$ using any permutation $\pi:V\to V$ is proportional to the number of non-fixed-points of $\pi$. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs. We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph). We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors (with very weak parameters but with some additional natural features). We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.Comment: Slightly modified and revised version of a CCC 2021 paper that also appeared on ECCC 27: 149 (2020