Extremal graphs without 4-cycles

We prove an upper bound for the number of edges a C4-free graph on q 2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q. Let n be a positive integer and G a graph. We define ex(n,G) to be the largest number of edges possible in a graph on n vertices that does not contain G as a subgraph; we call a graph on n vertices extremal if it has ex(n,G) edges and does not contain G as a subgraph. EX(n,G) is the set of all extremal G-free graphs on n vertices. The problem of determining ex(n,G) (and EX(n,G)) for general n and G belongs to an area of graph theory called extremal graph theory. Extremal graph theory officially began with Turán’s theorem that solves EX(n,Km) for all n and m, a result that is striking in its precision. In general, however, exact results for ex(n,G) (and especially EX(n,G)) are very rare; most results are upper or lower bounds and asymptotic results. For many bipartite G there is a large gap between upper and lower bounds. The question of ex(n,C4) (where C4 is a cycle of length 4) has an interesting history; Erdős originally posed the problem in 1938, and the bipartite version of this problem was solved by Reiman using a construction derived from the projective plane (see [3] and the references therein for a more detailed history). Reiman also determined the upper bound ex(n,C4) ≤ n4 (1 + √ 4n− 3) for general graphs, but this is known not to be sharp [11]. Erdős, Rényi, and Sós later showed that this is asymptotically correct using a construction known as the Erdős-Rényi graph derived from the orthogonal polarity graph of the classical projective plane [5] [6]. This is part of a more general family of graphs which we define below. Let π be a finite projective plane with point set P and line set L. A polarity φ of π is an involutionary permutation of P ∪ L which maps points to lines and lines to points and reverses containment. We call points absolute when they are contained in their own polar image. A polarity is called orthogonal if there are exactly q+1 absolute points. We define the polarity graph of π to be the graph with vertex set P , with two distinct vertices x, y adjacent whenever x ∈ φ(y). The graph is called an orthogonal polarity graph if the polarity is orthogonal. This graph is C4-free, has q 2 vertices of degree q+ 1, and q + 1 of degree q, for a total of 12q(q + 1) 2 edges. Füredi determined the first exact result that encompasses infinitely many n, namely that for q > 13 we have ex(q + q + 1, C4) ≤ 1 2q(q + 1) [7] [8], with equality if and only if the graph is an orthogonal polarity graph of a plane of order q. In particular, this shows ex(q + q + 1, C4) = 1 2q(q + 1) 2 for all prime powers q. The question of finding ex(n,C4) exactly for general n appears to be a difficult problem. Computer searches by Clapham et al. [4] and Yuansheng and Rowlinson [12] determined EX(n,C4) for all n ≤ 31. More general lower bounds are given in [1] by deleting carefully chosen vertices from the Erdős-Rényi graph. It is not known if any of these bounds are sharp in general. In particular it is not even known whether deleting a single vertex of degree q from an orthogonal polarity graph graph yields a graph which is still extremal, a question posed by Lazebnik in 2003 [10]. More generally, is ex(q + q, C4) ≤ 12q(q + 1) − q? In this paper we will prove the following theorem: Theorem 1. For q even, ex(q + q, C4) ≤ 12q(q + 1) − q. Preprint submitted to Elsevier January 25, 2012 It follows that equality holds for all q which are powers of 2. The question of determining EX(q+ q, C4) in this case is subtler; the searches referred to above showed that there are multiple constructions that achieve the bound for q = 2, 3, but for q = 4, 5 there is only one. In a subsequent paper, we will prove the following: Theorem 2. For all but finitely many even q, any C4-free graph with ex(q 2 + q, C4) edges is derived from an orthogonal polarity graph by removing a vertex of minimum degree. The proof of this result is much more lengthy and complicated than that of the inequality in Theorem 1, and requires q to be sufficiently large. The purpose of this paper is to give a simpler proof of the inequality and show it holds for all even q. We start with some notation. We let Xk be the set of vertices of degree k, X≤k be the set of vertices of degree at most k, E0 be 1 2q(q + 1) 2 − q, and n be the number of vertices (q + q). We will use Γ(x) to represent the vertices in the neighborhood of x. For our various lemmas, we will specify in each case whether q is an even number or simply a positive integer; however, in all cases we consider q ≥ 6. (We know from [4] and [12] that the inequality in Theorem 1 is true for q ≤ 5.) In general, we proceed indirectly. We will show that no C4-free graph with E0 + 1 edges can exist, from which we conclude that a graph cannot have more than that number of edges (as it would contain an impossible subgraph). We will use and generalize the techniques found in [7], [8], and [9]. Lemma 1. Let q be a natural number greater than 2 and let G be a C4-free graph on q 2 + q vertices with at least E0 edges. Then the maximum degree of a vertex in G is at most q + 2. Proof. Let u be a vertex of G of maximum degree d. Let e be the number of edges of G, e ≥ 1 2q(q+1)−q. We proceed by bounding the number of 2-paths in G which have no endpoints in Γ(v). This gives us: ( n− d 2 )