Let PG(q) denote the number of proper q-colorings of a graph G. This function, called the chromatic polynomial of G, was introduced by Birkhoff in 1912, who sought to attack the famous four-color problem by minimizing PG(4) over all planar graphs G. Since then, motivated by a variety of applications, much research was done on minimizing or maximizing PG(q) over various families of graphs. In this paper, we study an old problem of Linial and Wilf, to find the graphs with n vertices and m edges which maximize the number of q-colorings. We provide the first approach that enables one to solve this problem for many nontrivial ranges of parameters. Using our machinery, we show that for each q 4 and sufficiently large m < κqn 2 , where κq ≈ 1/(q log q), the extremal graphs are complete bipartite graphs minus the edges of a star, plus isolated vertices. Moreover, for q = 3, we establish the structure of optimal graphs for all large m n 2 /4, confirming (in a stronger form) a conjecture of Lazebnik from 1989.
It is shown that, for any lattice polytope PaW 1 , the set int {P)nlZ d (provided that it is non-empty) contains a point whose coefficient of asymmetry with respect to P is at most Sd-(8/+1) 2 ' * . If, moreover, P is a simplex, then this bound can be improved to 8 ( 8 / + I ) 2 . As an application, new upper bounds on the volume of a lattice polytope are deduced, given its dimension and the number of sublattice points in its interior. §1. Introduction. A lattice polytope in W 1 is a convex polytope whose vertices are lattice points, that is, points in ~L d . For an integer /2=1, let
Let Σ k consist of all k-graphs with three edges D1, D2, D3 such that |D1 ∩D2| = k −1 and D1 D2 ⊆ D3. The exact value of the Turán function ex(n, Σ k ) was computed for k = 3 by Bollobás [Discrete Math. 8 (1974), 21-24] and for k = 4 by Sidorenko [Math Notes 41 (1987), 247-259]. Let the k-graph T k ∈ Σ k have edges {1, . . . , k}, {1, 2, . . . , k − 1, k + 1}, and {k, k + 1, . . . , 2k − 1}. Frankl and Füredi [J. Combin. Theory Ser. (A) 52 (1989), 129-147] conjectured that there is n0 = n0(k) such that ex(n, T k ) = ex(n, Σ k ) for all n ≥ n0 and had previously proved this for k = 3 in [Combinatorica 3 (1983), 341-349]. Here we settle the case k = 4 of the conjecture.
We say that a first order formula Φ distinguishes a graph G from another graph G ′ if Φ is true on G and false on G ′ . Provided G and G ′ are non-isomorphic, let D(G, G ′ ) denote the minimal quantifier rank of a such formula. Let n denote the order of G. We prove that, if G ′ has the same order, then D(G, G ′ ) ≤ (n+3)/2. This bound is tight up to an additive constant of 1. Furthermore, we prove that non-isomorphic G and G ′ of order n are distinguishable by an existential formula of quantifier rank at most (n + 5)/2. As a consequence of the first result, we obtain an upper bound of (n+1)/2 for the optimum dimension of the Weisfeiler-Lehman graph canonization algorithm, whose worst case value is known to be linear in n.We say that a first order formula Φ defines a graph G if Φ distinguishes G from every non-isomorphic graph G ′ . Let D(G) be the minimal quantifier rank of a formula defining G. As it is well known, D(G) ≤ n + 1 and this bound is generally best possible. Nevertheless, we here show that there is a class C of graphs of simple, easily recognizable structure such that • D(G) ≤ (n + 5)/2 with the exception of all graphs in C;• if G ∈ C, then it is easy to compute the exact value of D(G).Moreover, the defining formulas in this result have only one quantifier alternation. The bound for D(G) can be improved for graphs with bounded vertex degrees: For each d ≥ 2 there is a constant c d < 1/2 such that D(G) ≤ c d n + O(d 2 ) for any graph G with no isolated vertices and edges whose maximum degree is d.Finally, we extend our results over directed graphs, more generally, over arbitrary structures with maximum relation arity 2, and over k-uniform hypergraphs.
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