A counterexample to a conjecture of Lovasz on the -coloring

Associated with every graph G of chromatic number is another graph G . The vertex set of G consists of all -colorings of G, and two -colorings are adjacent when they dier on exactly one vertex. According to a conjecture of Lovasz, this graph G must be disconnected. In this note we give a counterexample to this conjecture. In this paper we refute a conjecture of Lovasz. A complete account with the necessary background, and a proof of certain other instances of this conjecture can be found in [1]. For the purpose of the present note, no background is needed beyond elementary graph theory

Current Research

I am interested in various problems in extremal graph theory, expander graphs, coding theory, and computational complexity. My favorite tools are algebraic methods, spectral analysis, Markov chains, and the probabilistic method. I am currently studying the following problems: 1. Extremal problems on non-regular graphs (a) Perhaps the most important topic in extremal graph theory, is to determine the Turán number of a graph H. This is the maximal number of edges a graph of size n can have, without having a subgraph isomorphic to H. When H is non-bipartite, the asymptotics of this function is determined by the celebrated Erdös-Stone theorem, up to an 1 + o(1) factor. However, it is a longstanding problem to find good estimates on the Turán number of bipartite graphs, and in particular of even length cycles, C2k. In my past work in [1, 3, 4], I developed a technique for obtainin

universal cover

lower bound on the spectral radius of th