Some locally Kneser graphs

The Kneser graph $K(n,d)$ is the graph on the $d$-subsets of an $n$-set, adjacent when disjoint. Clearly, $K(n+d,d)$ is locally $K(n,d)$. Hall showed for $n \ge 3d+1$ that there are no further examples. Here we give other examples of locally $K(n,d)$ graphs for $n = 3d$, and some further sporadic examples. It follows that Hall's bound is best possible

Locally grid graphs: classification and Tutte uniqueness

We define a locally grid graph as a graph in which the structure around each vertex is a 3×3 grid ⊞, the canonical examples being the toroidal grids Cp×Cq. The paper contains two main results. First, we give a complete classification of locally grid graphs, showing that each of them has a natural embedding in the torus or in the Klein bottle. Secondly, as a continuation of the research initiated in (On graphs determined by their Tutte polynomials, Graphs Combin., to appear), we prove that Cp×Cq is uniquely determined by its Tutte polynomial, for p,q⩾6

ON THE STRUCTURE OF GRAPHS WHICH ARE LOCALLY INDISTINGUISHABLE FROM A LATTICE

license: © The Author(s) 2016 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited

A machine learning approach to constructing Ramsey graphs leads to the Trahtenbrot-Zykov problem.

Attempts at approaching the well-known and difficult problem of constructing Ramsey graphs via machine learning lead to another difficult problem posed by Zykov in 1963 (now commonly referred to as the Trahtenbrot-Zykov problem): For which graphs F does there exist some graph G such that the neighborhood of every vertex in G induces a subgraph isomorphic to F? Chapter 1 provides a brief introduction to graph theory. Chapter 2 introduces Ramsey theory for graphs. Chapter 3 details a reinforcement learning implementation for Ramsey graph construction. The implementation is based on board game software, specifically the AlphaZero program and its success learning to play games from scratch. The chapter ends with a description of how computing challenges naturally shifted the project towards the Trahtenbrot-Zykov problem. Chapter 3 also includes recommendations for continuing the project and attempting to overcome these challenges. Chapter 4 defines the Trahtenbrot-Zykov problem and outlines its history, including proofs of results omitted from their original papers. This chapter also contains a program for constructing graphs with all neighborhood-induced subgraphs isomorphic to a given graph F. The end of Chapter 4 presents constructions from the program when F is a Ramsey graph. Constructing such graphs is a non-trivial task, as Bulitko proved in 1973 that the Trahtenbrot-Zykov problem is undecidable. Chapter 5 is a translation from Russian to English of this famous result, a proof not previously available in English. Chapter 6 introduces Cayley graphs and their relationship to the Trahtenbrot-Zykov problem. The chapter ends with constructions of Cayley graphs Γ in which the neighborhood of every vertex of Γ induces a subgraph isomorphic to a given Ramsey graph, which leads to a conjecture regarding the unique extremal Ramsey(4, 4) graph

On graphs with a constant link, II

We study the following problem: For which finite graphs L do there exist graphs G such that the link (i.e., the neighborhood subgraph) of each vertex of G is isomorphic to L? We give a complete solution for the cases (i) L is a disjoint union of arcs, (ii) L is a tree with only one vertex of degree greater than two, (iii) L is a circle of prescribed length. Some other cases are also discussed. An interesting case is whether the situation is changed if we require G also to be finite. It transpires (see for example, Corollaries VII.3 and VII.4) that this is indeed the case.Part I of this paper will appear in [3]. It provides the basic definitions used in both part I and part II. Section III provides the basic tool, an identification procedure, that is used throughout the rest of the paper. Section IV sets up the basic building technique for the construction of more complicated graphs. It is shown how to build graphs such that the link of each vertex is an arc (of non-constant length), and how to control the proportional number of vertices with links of various lengths.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/22171/1/0000602.pd