@MISC{Chandran13bipartitepowers, author = {L. Sunil Chandran and Rogers Mathew}, title = {Bipartite Powers of k-chordal Graphs}, year = {2013} }

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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 dG(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.