The independence ratio of regular graphs

Abstract

set I of vertices of a graph G = (V, E) is said to be independent if no two vertices of I are joined by an edge of G. The maximal number of vertices in an independent set is the (vertex) independence number 1o(G) and /30(G)/IGI is the independence ratio of G. (Here I GI denotes the order of G, that is, the number of vertices of G. For the notation and general background, see [3].) The independence ratio is of great interest, least because of its close relationship to the chromatic number. An immediate corollary of a classical theorem of Brooks [6] is that if a graph has maximal degree = /\(G) and does contain a KA+1, a complete graph of order A + 1, then its independence ratio is at least 1/A. The condition that G does contain a KA+1 is clearly necessary; it is about the weakest condition expressing the fact that G is not too dense. Lately several authors have given lower bounds for the independence ratio under other conditions (see [1], [2], [8]-[11]). The sparseness of a graph is usually measured by its girth, the minimal length of a cycle. Let i(/, g) be the infimum of the independence ratio of graphs with maximum degree and girth at least g. In this notation Staton [9] proved that i(A, 4) > 5/(5A 1), and so in view of an example of Fajtlowicz [8] we have i(3, 4) = i(3, 5) = 5/14. Furthermore, concerning cubic graphs of large girth, Hopkins and Staton [10] showed that limgO, i(3, g) > 7/18. Our aim is to give upper bounds for i(A, g). In particular, we shall show that i(3, g) is bounded away from 2 which seemed the natural value for limg,. i(3, g) and, in fact, i(3, g) < 6/13 for every g. In order to do this one has to show that there are cubic graphs of large girth without many independent vertices. We shall construct such graphs but show their existence by probabilistic methods. Probabilistic methods have been more and more in use since Erdos [7] tackled a similar problem over twenty years ago. However, the proof of our main result has become possible because of a very recent probabilistic model of regular graphs [5]. We state the theorem in its sharpest, though rather unattractive, form; later we deduce from it some more appealing and slightly weaker results.