Induced matchings in bipartite graphs

All graphs in this paper are understood to be finite, undirected, without loops or multiple edges. The graph G' = (V', E') is called an induced subgraph of G = (V, E) if V' ~ V and uv E E' if and only if {u, v} ~ V', uv E E. The following two problems about induced matchings have been formulated by Erdos and Nesetril at a seminar in Prague at the end of 1985: 1. Determine f(k, d), the maximum number of edges in a graph which has maximum degree d and contains no induced (k + 1)-matching (an induced matching of k + 1 edges). Fork= 1 this was asked earlier by Bermond, Bond and Peyrat (see [1]). 2. Let q*(G) denote the minimum integer t for which the edge set of G can be partitioned into t induced matchings of G. (We will call q*(G) the strong chromatic index of G.) As is done in Vizing's theorem, find the best upper bound of q * (G) when G has maximum degree d. It was shown in [1] that (for d even) /(1, d) d id and the extremal graph is unique (each vertex of a five cycle is multiplied by d/2). This result suggests that f(k, d) =idk. Perhaps a stronger conjecture is also true, namely, that q*(G)::::; id when G has maximum degree d. In this paper the analogous extremal problem for bipartite graphs is considered. It is shown that bipartite graphs of maximum degree d without an induced (k + 1)-matching ha~e at most kd edges (Theorem 1). Extremal graphs for k > 1 are not unique but can be completely described (Theorem 2). It is also shown (Theorem 3) that when the extremal problem is restricted to connected bipartite graphs, the extremal number drops by at least d (if k > 2). We