2 research outputs found

    Almost All Graphs With 2.522n Edges Are Not 3-Colorable

    No full text
    We show that if a random graph on n vertices has m = (r + o(1))n edges, where r=2.522, then almost surely it is not 3-colorable. The previous best such value for r was 2.571[5]. Our result follows from applying ideas of [4] to the k-coloring problem. The development of the proof is for k-colorings, where k is an arbitrary constant, and we present similar improvements for small values of k. 1 Introduction The term "almost all" in the title has the meaning introduced by Erdos and R'enyi [3]. If N(n; m; A) stands for the number of graphs with vertices f1; : : : ; ng, precisely m edges, and some property A, saying that almost all graphs with n vertices and m(n) edges have property A, means that lim n!1 N(n; m(n); A) \Gamma ( n 2 ) m(n) \Delta = 1 : (In the terminology of Bollob'as [1], we are using Model GM of a random graph.) 2 Upper bounds for k-colorability 2.1 The first moment method In the rest of the paper a random graph G(V; E) will always have m = m(n) = rn edges, for ..
    corecore