Bipartite Powers of k-chordal Graphs

Let k be an integer and k >= 3. A graph G is k-chordal if G does not have an induced cycle of length greater than k. From the definition it is clear that 3-chordal graphs are precisely the class of chordal graphs. Duchet proved that, for every positive integer m, if G m is chordal then so is G(m+2). Brandst " adt et al. in [Andreas Brandsadt, Van Bang Le, and Thomas Szymczak. Duchet- type theorems for powers of HHD- free graphs. Discrete Mathematics, 177(1- 3): 9- 16, 1997.] showed that if G m is k - chordal, then so is G(m+2). Powering a bipartite graph does not preserve its bipartitedness. In order to preserve the bipartitedness of a bipartite graph while powering Chandran et al. introduced the notion of bipartite powering. This notion was introduced to aid their study of boxicity of chordal bipartite graphs. The m - th bipartite power G([m]) of a bipartite graph G is the bipartite graph obtained from G by adding edges (u; v) where d G (u; v) is odd and less than or equal to m. Note that G([m]) = G([m+1]) for each odd m. In this paper we show that, given a bipartite graph G, if G is k-chordal then so is G [m], where k, m are positive integers with k >= 4