A new lower bound on the independence number of a graph

1 A new lower bound on the independence number of a graph O.Kettani Faculté des sciences et techniques-Mohammedia-Maroc Abstract For a given connected graph G on n vertices and m edges, we prove that its independence number α (G) is at least ( (2m+n+2) -((2m+n+2) 2 -16n 2 ) ½ ) /8. Intoduction Let G=(V,E) be a connected graph G on n= │ V │ vertices and m= │ E │ edges. For a subgraph H of G and for a vertex i ∈ V(H), let d H (i) be the degree of i in H and let N H (i) be its neighbourhood in H. Let δ (H) and ∆ (H) be the minimum degree and the maximum degree of H, respectivly. A subse t X of V is called independent if its vertices are mutually non-adjacent. The independen ce number α (G) is the largest cardinality among all independent sets of G. The problem of finding an independent set of maximum cardinality is know to be NP –complete[1]. Some approximation algorithms was designed to tackle this problem, among them, the well know MIN algorithm [4], which can be imple mented in time linear in n and m : G 1 :=G, j :=1 While V(G j ) ≠ ∅ do Begin Choose i j ∈ V(G j ) with d Gj (i j )= δ (G j ), delete {i j } ∪ N Gj (i j ) to obtain G j+1 and set j :=j+1 ; End ; k :=j-1 stop. Let k MIN be the smallest k t he algorithm MIN provides for a given connected graph G. Harant [3] proved that α (G) ≥ k MIN ≥ ((2m+n+1) -((2m+n+1) 2 -4n 2 ) ½ )/2. The purpose of the presen t note is to improve this lower bound. Claim : For a given connected graph G on n vertices and m edges, α (G) ≥ ( (2m+n+2) -((2m+n+2) 2 -16n 2 ) ½ ) /8. The proof starts with the inequality (1) proved by Harant [3] : k MIN ≥ n 2 / ( 2m+n- ∑ (d G (i)- δ (G j )) ) (1) i ∈ I n and uses a variation of the one given by Halldorson [2]. 2 For j=1,…, k MIN , let d Gj (i j ) be the degree in the remaining graph of the j–th vertice choosed at the j–th iteration of the algorithm MIN. The number of v ertices deleted in the j–th iteration is thus 1+ d Gj (i j ) and the sum of the degrees of the 1+ d Gj (i j ) vertices deleted is at least (1+ d Gj (i j ))d Gj (i j ). Thus the number of edges removed in the j–th iteration is at least (1+ d Gj (i j ))d Gj (i j )/2. Let X be an independent set of G of maximum cardinality α , and let k j be the number of vertices among the 1+ d Gj (i j ) vertices deleted in the j–th iteration that are also contained in X. k MIN Then ∑ k j = α j=1 Since X is edgless, and G is connected then the nu mber of edges removed in the j–th iteration (j=1,…, k MIN -1) is at least :  1+ d Gj (i j )  +  k j  +1  2   2  (for j=1 ,…, k MIN -1 , there is at least one edge between N Gj (i j ) and G j+1 , because G is supposed connected). In the k MIN –th iteration, at least  1+ d Gj (i kMIN )  +  k kMIN   2   2  edges are removed Hence we obtain the following inequality : k MIN -1 m ≥ ∑ (  1+ d Gj (i j )  +  k j  +1 ) +  1+ d Gj (i kMIN )  +  k kMIN  j=1  2   2   2   2  then : k MIN k MIN k MIN 2m ≥ 2k MIN -2+ ∑ ( ( 1+ d Gj (i j ) ) d Gj (i j ) ) + ∑ k j + ∑ (k j ) 2 j=1 j=1 j=1 3 consequently : k MIN 2m ≥ 4k MIN -2 + ∑ ( (1+ d Gj (i j )) d Gj (i j ) ) (2 ) j=1 On the other hand : Since ∀ (j,j’) ∈ {1 ,…, k MIN }, j ≠ j’ ⇒ ( {i j } ∪ N Gj (i j ) ) ∩ ( {i j’ } ∪ N Gj’ (i j’ ) ) = ∅ and k MIN I n = ∪ ( {i j } ∪ N Gj (i j ) ) ={1,…,n} j=1 then k MIN ∑ δ (G j )= ∑ ∑ δ (G j ) i ∈ I n j=1 i ∈ {i j } ∪ N Gj (i j ) k MIN k MIN ∑ δ (G j )= ∑ (1+ d Gj (i j )) δ (G j ) ≤ ∑ ( (1+ d Gj (i j ))d Gj (i j ) ) i ∈ I n j=1 j=1 thus k MIN ∑ (d G (i)- δ (G j ))=2m- ∑ δ (G j ) ≥ 2m- ∑ ( (1+ d Gj (i j ))d Gj (i j ) ) i ∈ I n i ∈ I n j=1 by using inequality (2) we get : ∑ (d G (i)- δ (G j )) ≥ 4k MIN -2 i ∈ I n then inequality (1) implies : k MIN ≥ n 2 /(2m+n+2-4k MIN ) and consequently : k MIN ≥ ( (2m+n+2) -((2m+n+2) 2 -16n 2 ) ½ ) /8. 4 Conclusion This note presented an improved lower bound on the independence numbe r of a graph, and as a future work, our intention is to prove that this bound is optimale for an important class of graphs. References : [1]. M. R. Garey, D. S. Johnson, "Computers and intractability. A guide to the theory of NP-completness".1979 [2]. Halldorson, Radhakrishnan. Geed is good : Approximating independent sets in sparse and bounded degree graphs. Algorithmica, 1996. [3]. J. Harant:. T.. I. Schiermeye r On the independence number of a graph in terms of order and size. Discrete Math. 232 (2001 ) 131-138 [4]. O.Murphy:. Lower bounds on the stability number of graphs computed in terms of degrees. Discrete Math. 90 (1991) 20 7-211