Maximum number of colorings of (2k, k²)‐graphs

Abstract

Let F 2k,k 2 consist of all simple graphs on 2k vertices and k 2 edges. For a simple graph G and a positive integer λ, let P G (λ) denote the number of proper vertex colorings of G in at most λ colors, and let f (2k,and K k,k is the only extremal graph. We also prove that f (2k, k 2 , 4) = (6 + o(1))4 k as k → ∞.

Extremes of the internal energy of the Potts model on cubic graphs

“…Linial asked which graph on n vertices with m edges maximizes . After a series of bounds by Labeznik and coauthors , Loh, Pikhurko, and Sudakov gave a complete answer to this question for a wide range of parameters q , n , m , using the Szemerédi's Regularity Lemma to reduce the maximization problem over graphs to a quadratic program in variables.…”

An Extremal Property of Turán Graphs, II

“…Then, if G is a (4, 4)-graph nonisomorphic to T 2 (4), G must consist of a triangle with an extra edge attached to one vertex. In that case, P G (4) = 4 · 3 · 2 · 3 < P T 2 (4) (4), and so we are done.…”

“…Then, the graph G = G − x is a (4, 4)-graph, and therefore P G (4) ≤ P T 2 (4) (4) = 84. Since every coloring of G can be extended to G by assigning a color to x, and x is part of a triangle, therefore P G (4) = 2P G (4) ≤ 2P T 2 (4) (4) < P T 2 (5) (4). If no such vertex of degree 2 exists, then G must be isomorphic to K 4 plus an isolated vertex, and therefore P G (4) = 24 · 4 = 96 < P T 2 (5) , as desired.…”

“…Here, we will mention only those results which are most relevant to this article. For λ = 3 and n = 2k, the conjecture was established in [4]. The result was extended to (r + 1)-colorings of (n, t r (n))-graphs for r ≥ 2 and large n in [7], and to (r + 1)colorings of (n, t r (n))-graphs for r ≥ 2 and all n ≥ r in [5].…”

“…If G is nonisomorphic to T 2 (n − 1), then by deleting e(G ) − t 2 (n − 1) edges from G , we obtain a subgraph G which is a (n − 1, t 2 (n − 1))-graph that contains a triangle. Obviously, P G (4) ≤ P G (4). As every coloring of G can be extended to a coloring of G by assigning a color to x in at most two ways (as x is joint to the endpoints of an edge yz of G ), we have by induction…”