Abstract
Graph Theory 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 Gm is chordal then so is Gm+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 Gm is k-chordal, then so is Gm+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.
Highlights
A hole is a chordless cycle in a graph
A graph is k-chordal if it has no holes with more than k vertices in it
Chordal graphs are exactly the class of 3-chordal graphs and chordal bipartite graphs are bipartite, 4-chordal graphs. k-chordal graphs have been studied in the literature in
Summary
A hole is a chordless (or an induced) cycle in a graph. The chordality of a graph G, denoted by C(G), is defined to be the size of a largest hole in G, if there exists a cycle in G. Paulraja [1] proved that odd powers of chordal graphs are chordal. Note that G[m] = G[m+1] for each odd m It was shown in [4] that the m-th bipartite power of a tree is chordal bipartite. The fact that the chordal bipartite graph under consideration was obtained as a bipartite power of a tree was crucial for proving that its boxicity was high. Since trees are a subclass of chordal bipartite graphs, a natural question that came up was the following: is it true that the m-th bipartite power of every chordal bipartite graph is chordal bipartite? Note that the special case when k = 4 gives us the following result: chordal bipartite graphs are closed under bipartite powering
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