### Bipartite Powers of k-chordal Graphs

*L. Sunil Chandran, Rogers Mathew*

#### Abstract

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}. Brandstädt et al. in [Andreas Brandstädt, 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.Full Text: PDF PostScript