Bipartite powers of k-chordal graphs

TL;DR

This paper proves that the bipartite power of a k-chordal bipartite graph remains k-chordal for k ≥ 4 and odd powers, extending known properties of graph powers to bipartite graphs.

Contribution

It establishes that bipartite powers preserve the k-chordal property for k ≥ 4 and odd powers, a significant extension of existing graph power theorems.

Findings

Bipartite powers of k-chordal graphs are also k-chordal for k ≥ 4.

The result applies specifically to odd powers of bipartite graphs.

This extends the understanding of power preservation in bipartite graph classes.

Abstract

Let k be an integer and k \geq 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 Brandst\"adt, 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. Given a bipartite graph G and an odd positive integer m, we define the graph G^{[m]} to be a bipartite graph with V(G^{[m]})=V(G) and E(G^{[m]})={(u,v) | u,v \in V(G), d_G(u,v) is odd, and d_G(u,v) \leq m}. The graph G^{[m]} is called the m-th bipartite power of G. 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 such that k \geq 4 and m is odd.