On the number of edges of a graph and its complement

Let G ¼ ð V ; E Þ be a graph. The complement of G is the graph G : ¼ ð V ; ½ V (cid:2) 2 n E Þ where ½ V (cid:2) 2 is the set of pairs f x ; y g ofdistinctelementsof V .If K isasubsetof V ,the restrictionof G to K isthe graph G ↾ K : ¼ ð K ; ½ K (cid:2) 2 ∩ E Þ . Weprovethatif G ¼ ð V ; E Þ isagraphand k isaninteger,2 ≤ k ≤ v − 2,thenthereisa k -elementsubset K of V such that e ð G ↾ K Þ ≠ e ð G ↾ K Þ , moreover the condition k ≤ v − 2 is optimal. We also study the case e ð G ↾ K Þ u e ð G ↾ K Þð mod p Þ where p is a prime number. Following a question from M.Pouzet, we show this: Let G ¼ ð V ; E Þ beagraphwith v vertices.If e ð G Þ ≠ e ð G Þ (resp. e ð G Þ ¼ e ð G Þ )thenthereisanincreasingfamily ð H n Þ 2 ≤ n ≤ v (resp. ð H n Þ 2 ≤ n ≤ v − 2 ) of n -element subsets H n of V such that e ð G ↾ H n Þ ≠ e ð G ↾ H n Þ for all n . Similarly if e ð G Þ u e ð G Þ ð mod p Þ where p is a prime number, p > v − 2, then there is an increasing family ð H n Þ v of n -element subsets H n of V such that e ð G ↾ H n Þ u e ð G ↾ H n Þð mod p Þ for all integer n ∈ f 2 ; 3 ; ... ; v g .


Introduction
Our notations and terminology follow [2]. A graph is an ordered pair G :¼ ðV ; EÞ (or ðV ðGÞ; EðGÞÞ), where E is a subset of ½V 2 , the set of pairs fx; yg of distinct elements of V. Elements of V are the vertices of G and elements of E are its edges. An edge fx; yg is also noted by x y. The cardinality jV j of V is called the order of G. Two distinct vertices x and y are adjacent if x y ∈ EðGÞ, otherwise x and y are non-adjacent. We denote by eðGÞ :¼ jEðGÞj the number of edges of G. The degree of a vertex x of G, denoted by d G ðxÞ, is the number of edges which contain x. The graph G is δ -regular (or regular) if d G ðxÞ ¼ δ for all x ∈ V; Edges of a graph and its complement δ is called the degree of the regular graph G. The complement of G is the graph G :¼ ðV ; ½V 2 n EÞ. If K is a subset of V, the restriction of G to K, also called the induced subgraph of G on K, is the graph G ↾K :¼ ðK; ½K 2 ∩ EÞ. For instance, given a set V, ðV ; 0 =Þ is the empty graph on V whereas ðV ; fxy : x ≠ y ∈ V gÞ is the complete graph.
Our first result is Theorem 1.1, we prove that: given a graph G ¼ ðV ; EÞ and k be an integer, 2 ≤ k ≤ v − 2, we cannot have eðG ↾K Þ ¼ eðG ↾K Þ for all k -element subsets K of V, moreover the condition k ≤ v − 2 is optimal, indeed for k ¼ v − 1 a counterexample is given by (2) of Theorem 1.1.
(3) Let p be a prime number with p ≥ 3 such that 2d G ðxÞ ≡ v − 1ðmod pÞ for all x ∈ V .
Our second result is Theorem 1.2. Given a graph G ¼ ðV ; EÞ, p a prime number, and k an integer, 2 ≤ k ≤ v − 2, under some conditions on k, we cannot have eðG ↾K Þ ≡ eðG ↾K Þ ðmod pÞ for all k-element subsets K of V. (1) If ( p ¼ 2 and k ≡ 2ðmod 4Þ ) or ( p ≥ 3 and k u 0; 1ðmod pÞ ), then there is a k-element subset K of V such that eðG ↾K Þ u eðG ↾K Þðmod pÞ.
(2) If p ≥ 3 and k ≡ 0ðmod pÞ then there is a k-element subset K of V such that eðG ↾K Þ u eðG ↾K Þðmod pÞ if and only if G is neither the complete graph nor the empty graph.
Our third result is Theorem 1.3. It is related to a question that M.Pouzet asked us about the existence, in a graph G ¼ ðV ; EÞ, of an increasing family ðH n Þ n of n -element subsets H n of V such that eðG ↾Hn Þ ≠ eðG ↾Hn Þ.
(4) Let p be a prime number, p > v − 2 . If eðGÞ u eðGÞðmod pÞ then there is an increasing family ðH n Þ 2≤n≤v of n-element subsets H n of V such that eðG ↾H n Þ u eðG ↾H n Þðmod pÞ for all integer n ∈ f2; 3; . . . ; vg.

Incidence matrices
We consider the matrix W t k defined as follows: Let V be a finite set, with v elements. Given non-negative integers t; k, let W t k be the v t by v k matrix of 0's and 1's, the rows of which are indexed by the t-element subsets T of V, the columns are indexed by the k-element subsets K of V, and where the entry W tk ðT; KÞ is 1. T ⊆ K and is 0 otherwise. The matrix transpose of W t k is denoted t W t k . A fundamental result, due to D.H. Gottlieb [4], and independently W. Kantor [5], is this: [4], W. Kantor [5]). For t ≤ minðk; v − kÞ , W t k has full row rank over the field ℚ of rational numbers.
then, from Theorem 2.1, we have the following result.
Corollary 2.2. For t ≤ minðk; v − kÞ , the rank of W t k over the field ℚ of rational numbers is v t and thus Kerð t W t k Þ ¼ f0g.
Corollary 2.2 and the following theorem are important tools in the proof of our main results. In fact, Theorem 2.3 has made to establish a version modulo a prime of Kelly's combinatorial lemma [6]; it also allows to obtain a version modulo a prime of the particular version of Pouzet's combinatorial lemma [7]. Let n; p be positive integers, the decomposition of n ¼ P nðpÞ i¼0 n i p i in the basis p is also denoted by ½n 0 ; n 1 ; . . . ; n nðpÞ p where n nðpÞ ≠ 0 if and only if n ≠ 0. Theorem 2.3 [1]. Let p be a prime number. Let v; t and k be non-negative integers, k ¼ ½k 0 ; k 1 ; . . . ; k kðpÞ p , t ¼ ½t 0 ; t 1 ; . . . ; t tðpÞ p , t ≤ minðk; v − kÞ . We have: (1) k j ¼ t j for all j < tðpÞ and k tðpÞ ≥ t tðpÞ if and only if Kerð t W tk Þ ¼ f0g (mod p ).
(2) t ¼ t tðpÞ p tðpÞ and k ¼ P kðpÞ i¼tðpÞþ1 k i p i if and only if dKerð t W tk Þ ¼ 1 (mod p ) and fð1; 1; . . . ; 1Þg is a basis of Kerð t W t k Þ.

Theorem 2.4 (Lucas' Theorem
The following result is a consequence of Lucas' theorem.
In the two cases, Kerð t W t k Þ ¼ f0g (mod p). Assume that eðG ↾K Þ ≡ eðG ↾K Þðmod pÞ for all k-element subsets K of V. Then w G W 2k ¼ w G W 2 k ðmod pÞ. As Kerð t W t k Þ ¼ f0g (mod p), then w G ¼ w G ðmod pÞ, so G ¼ G, which is impossible. Then there is a k-element subset K of V such that eðG ↾K Þ u eðG ↾K Þðmod pÞ.
(2) If p ≥ 3 then t ¼ t 0 ¼ 2 ¼ t tðpÞ . Since k ≡ 0 ðmod pÞ then k 0 ¼ 0, and thus k ¼ P kðpÞ i¼tðpÞþ1 k i p i . By Theorem 2.3, fð1; 1; . . . ; 1Þg is a basis of Kerð t W t k Þ. If G is the complete graph or the empty graph then eðG ↾K Þ ≡ eðG ↾K Þðmod pÞ. Indeed, if G is the complete graph then eðG ↾K Þ ¼ Conversely if G is neither the complete graph nor the empty graph, assume that eðG ↾K Þ ≡ eðG ↾K Þðmod pÞ for all k-element subsets K of V. Then w G W 2 k ¼ w G W 2 k ðmod pÞ. So w G − w G ¼ λð1; . . . ; 1Þðmod pÞ with λ ∈ f0; 1; − 1g. As G is neither the complete graph nor the empty graph, there are i; j such that g i ¼ 0 and g j ¼ 1, so g i − g i ¼ −1 and g j − g j ¼ 1. Then λ ≠ 1 and λ ≠ − 1. Thus λ ¼ 0, and w G ¼ w G ðmod pÞ, so G ¼ G, which is impossible. Then there is a k-element subset K of V such that eðG ↾K Þ u eðG ↾K Þðmod pÞ. ,

Proof of Theorem 1.3
We need the following lemma. (1) Let k be an integer, k − 2 eðGÞ ðmod pÞ then there is a k-element subset K of V such that eðG ↾K Þ u eðG ↾K Þðmod pÞ .
Proof. (1) It is an immediate consequence of the following formula v À 2 k À 2 eðGÞ ¼ X (2) We make the proof by induction on v. We set H v :¼ V. We assume that H i is defined for all k ≤ i ≤ v. Let us define H k−1 . As eðG ↾H k Þ ≠ eðG ↾H k Þ then by (1), there is x ∈ H k such that eðG ↾H k − xÞ ≠ eðG ↾H k − xÞ. We set H k−1 :¼ H k n fxg. So H k−1 ⊂ H k and eðG ↾H k − 1 Þ ≠ eðG ↾H k − 1 Þ.