Technische Universität Ilmenau Institut für Mathematik

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1 Technische Universität Ilmenau Institut für Mathematik Preprint No. M 0/08 A lower bound on independence in terms of degrees Harant, Jochen August 00 Impressum: Hrsg.: Leiter des Instituts für Mathematik Weimarer Straße Ilmenau Tel.: Fax:

2 A Lower Bound on Independence in Terms of Degrees Jochen Harant Institut für Mathematik TU Ilmenau, Postfach D Ilmenau, Germany Abstract We prove a new lower bound on the independence number of a simple connected graph in terms of its degrees. Keywords: Independence; stability; connected graph AMS subject classification: 05C69 Introduction We consider finite, simple, and undirected graphs G with vertex set V. For a graph G, we denote its order by n and its size by m, respectively. The degree of u in G is denoted by du) and is the maximum degree of G. A set of vertices I V in a graph G is independent, if no two vertices in I are adjacent. The independence number α of G is the maximum cardinality of an independent set of G. The independence number is one of the most fundamental and well-studied graph parameters [6]. In view of its computational hardness [5] various bounds on the independence number have been proposed. The following classical bound holds for every graph G and is due to Caro and Wei [, 7] α du) +. ) Since the only graphs for which ) is best-possible are the disjoint unions of cliques, additional structural assumptions excluding these graphs allow improvements of ). A natural candidate for such assumptions is connectivity. For connected graphs, Harant and Rautenbach proved [] cf. also [3] and [4]) Theorem If G is a connected graph, then there exist a positive integer k N and a function φ : V N 0 with non-negative integer values such that φu) du) for all u V, α k du) + φu), ) and φu) k ). 3)

3 Note that Theorem is best-possible for the connected graphs which arise by adding bridges to disjoint unions of cliques, i.e. it is best-possible for the intuitively most natural candidate of a connected graph with small independence number. In [3], a weaker version of Theorem is proved. This result is obtained from Theorem by replacing the inequality 3) by φu) k. For an integer l with 0 l m let fl) = min du)+ φu), where the minimum is taken over all integers 0 φu) du) with φu) = l. Obviously, f is strictly increasing. With this function f, it follows the existence of positive integers k and k such that α k + fk ) put k = k and use the result in [3]) and α k + fk ) with k = k ) and Theorem ). After extending f to real arguments, in [4], it is proved that the function l + fl) is continuous and strictly increasing and that k is at least the unique zero k 0 of this function. Finally, α k 0 + is the main result in [4]. Here we will show that the continuous function l + fl) is also strictly increasing. If we assume that for the graph in question then f) > f0) =. It will be proved du)+ du)+ that there is a unique solution l 0 of the equation l + = fl) and, because l + fl) is strictly increasing and + f) < 0, it follows l 0 >. Consequently, l 0 + = fl 0 ) > f l 0 + ), since f is strictly increasing, hence, l 0 > k 0. The inequality α l 0 + is the content of the following Theorem. In case, Remark gives a lower bound on the improvement du)+ l 0 + ) k 0 + ) = fl 0 ) fk 0 ). Theorem Let G be a finite, simple, connected, and non-complete graph on n 3 vertices of size m n. Moreover, let α n be the independence number, be the maximum degree of G, n j be the number of vertices of degree j in G, and jj + ) xj) = jj + ) j + j))n j + )n j+ + n j j n ) for j {,..., }. Then i) there is a unique j 0 {,..., } such that 0 xj 0 ) < n n j0 and ii) α du) + ) + n + ) + n + n ) n n j0 + j 0 + )j 0 + ) + xj 0) j 0 + )j 0 = xj 0) + n j0 + + n j j 0 )n +. Proof of Theorem In the sequel let k be the lower bound on α of Theorem. By Theorem, it follows Lemma k fk )).

4 For a finite family F of integers let maxf ) be a maximum member of F. Note that a member of a family may occur more than once. If for instance F = {,, } then F \{maxf )}) {maxf ) } = {,, }. The following Lemma, Lemma 3, and Lemma 4 are proved in [3] and [4]. Lemma Given an integer l with 0 l m, the following algorithm calculates fl): Input: The family F = {du) u V }. j := 0, while j < l do begin F := F \ {maxf )}) {maxf ) }; j := j + end Output: fl) = m F m+. Lemma 3 is a consequence of Lemma. Lemma 3 Given an integer 0 l m, i) there are unique integers j and x with j {,..., } and x {0,..., n n j } such that l = n + n + n ) n + n n j+ ) + x = x + n j+ + n j j)n and ii) fl) = n n j x) j+ + x j + n j j n = n n j ) j+ + x jj+) + n j j n. By Lemma 3, it follows Lemma 4 If l = x + n j+ + n j j)n with j {,..., } and x {0,..., n n j } than fl + ) fl) = jj+). Using Lemma 3, the calculation of fl) is possible without taking a minimum and without using the algorithm above. We will now define the function f for real l [, m). For given j {,..., } and a real number x with 0 x < n n j let the real numbers l and fl) implicitly) be defined as l = x + n j+ + n j j)n and fl) = n n j ) j+ + x jj+) + n j j n. We will prove Lemma 5. Lemma 5 The function g with gl) = l + fl) is continuous and strictly increasing on [, n). Proof. Consider l [, n). Then there are j {,..., } and x with 0 x < n n j such that l = x + n j+ + n j j)n. If j = then n > l n + n )n = m n, a contradiction to n m. Hence, j, and l belongs to the interval Ij) = [n j+ + n j j)n, n j + n j j + )n ). By Lemma 3, gl + ɛ) gl) = ɛ jj+) ) and, consequently, gl) is continuous and, because j, strictly increasing on Ij). Note that Ij) Ij ) = if j j and that I)... I ) = [, m n) [, n). It is easy to see that g is also continuous in l = n j+ + n j j)n for j {3,..., }. Since the classical bound due to Caro and Wei is tight only for complete graphs, it follows g0) = < 0, and, by Lemma, gk )) 0. Using Lemma 5, there is a unique du)+ zero l 0 = xj 0 ) + n j0 + + n j j 0 )n of g with < l 0 k ) α ) < n and 0 xj 0 ) < n n j0. It follows Lemma 6. Lemma 6 α k l 0 +, where l 0 0, n] is the unique solution of l + = fl). 3

5 Considering the equation l 0 + = fl 0 ), i.e. + xj 0 ) + n j0 + + n j j 0 )n = n n j0 ) it follows x 0 = j 0j 0 +) j 0 j 0 +) We obtain Lemma 7. jj+) jj+) j xj 0) j 0 j 0 +) + n j 0 j 0 + j 0))n j 0 + )n j n j 0 j n ). Lemma 7 If j {,..., } and l = x + n j+ + n j j)n with 0 x < n n j, then l + = fl) if and only if x = j+ j))n j+ )n j+ + n j j n ). Now we complete the proof of Theorem. By Lemma 4 and Lemma 6, α k fl 0 ) = f0) + f) f0)) f l 0 ) f l 0 )) + fl 0 ) f l 0 )) = du)+ ) + n +) + n +n ) n +...+n j0 + j 0 +)j 0 +) + xj 0) j 0 +)j 0 because l 0 = xj 0 ) + n j0 + + n j j 0 )n = n + n + n ) n + n n j0 +) + xj 0 ). j n With fl 0 ) = l 0 + = xj 0)+n j0 ++n j j 0 )n + Theorem is proved. 3 Remarks The following Remark is proved in the introduction. Remark If du)+ then l 0 > k 0. Remark compares the lower bound l 0 + on α in Theorem to the lower bound k 0 + on α in the main result in [4]. Remark If du)+ and k 0 = n + n + n ) n + n n j+ ) + x with 0 x < n n j then l 0 k 0 k 0 jj+). Proof. Remark implies l 0 k 0 = fl 0 ) fk 0 ) > fk 0 ) fk 0 ). According to Lemma, the family F contains the member, the member,..., and the member exactly n times, n times,..., n times, respectively. Therefore, let the output fl) of the algorithm in Lemma be denoted by f n,...,n l). With this notation, for example f n,...,n ) = f n,...,n +,n 0). Using Lemma 3 ii), it follows f n,...,n k 0 ) = f n,...,n j,n j +x,n +...+n j x0) and f n,...,n k 0 ) = f n,...,n j,n j +x,n +...+n j xk 0 ). Consequently, f n,...,n k 0 )) f n,...,n k 0 ) = f n,...,n j,n j +x,n +...+n j xk 0 ) f n,...,n j,n j +x,n +...+n j x0) = f n,...,n j,n j +x,n +...+n j x)k 0 ) f n,...,n j,n j +x,n +...+n j x)k 0 )) +f n,...,n j,n j +x,n +...+n j x)k 0 ) f n,...,n j,n j +x,n +...+n j x)k 0 )) f n,...,n j,n j +x,n +...+n j x)) f n,...,n j,n j +x,n +...+n j x)0)). Note that the expressions f n,...,n j,n j +x,n +...+n j x)s) f n,...,n j,n j +x,n +...+n j x)s ) equal fractions of type aa+) see Lemma 3 and Lemma 4) with a j for s =,..., k 0. Thus, f n,...,n k 0 ) f n,...,n k 0 ) k 0 jj+). ) 4

6 For integers r and s, consider the graph G r,s obtained from s copies of the clique K r on r vertices and adding s mutually independent edges between these cliques such that G r,s is connected. It follows = r, n j = 0 for j < r, n r = sr s ), n r = s ), and α = s for G r,s. Using Theorem, we obtain xr ) = r )r r )r r )n r + n r r ) ) = r )r r )r sr 4s ) r )s ) + r = 0. Hence, j 0 = r, xj 0 )+n j0 ++n j j 0 )n + = nr + = s = α, and Remark 3 follows. Remark 3 There are infinitely many graphs G such that the lower bound on α of Theorem is tight. References [] Y. Caro, New Results on the Independence Number, Technical Report, Tel-Aviv University, 979. [] J. Harant and D. Rautenbach, Independence in Connected graphs, submitted to Discrete Math.. [3] J. Harant and I. Schiermeyer, On the independence number of a graph in terms of order and size, Discrete Math. 3 00), [4] J. Harant and I. Schiermeyer, A lower bound on the independence number of a graph in terms of degrees, Discuss. Math., Graph Theory 6 006), [5] J. Håstad, Clique is hard to approximate within n ɛ, in: Proceedings of 37th Annual Symposium on Foundations of Computer Science FOCS), 996, [6] V. Lozin and D. de Werra, Foreword: Special issue on stability in graphs and related topics, Discrete Appl. Math ), -. [7] V.K. Wei, A Lower Bound on the Stability Number of a Simple Graph, Technical memorandum, TM , Bell laboratories, 98. 5