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arXiv · 2311.06508

Resonance graphs of plane bipartite graphs as daisy cubes

Simon Brezovnik·Zhongyuan Che·Niko Tratnik·Petra Žigert Pleteršek
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Abstract

We characterize plane bipartite graphs whose resonance graphs are daisy cubes, and therefore generalize related results on resonance graphs of benzenoid graphs, catacondensed even ring systems, as well as 2-connected outerplane bipartite graphs. Firstly, we prove that if $G$ is a plane elementary bipartite graph other than $K_2$, then the resonance graph of $G$ is a daisy cube if and only if the Fries number of $G$ equals the number of finite faces of $G$. Next, we extend the above characterization from plane elementary bipartite graphs to plane bipartite graphs and show that the resonance graph of a plane bipartite graph $G$ is a daisy cube if and only if $G$ is weakly elementary bipartite such that each of its elementary component $G_i$ other than $K_2$ holds the property that the Fries number of $G_i$ equals the number of finite faces of $G_i$. Along the way, we provide a structural characterization for a plane elementary bipartite graph whose resonance graph is a daisy cube, and show that a Cartesian product graph is a daisy cube if and only if all of its nontrivial factors are daisy cubes.

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Simon Brezovnik, Zhongyuan Che, Niko Tratnik, Petra Žigert Pleteršek. 2024-12-24. Resonance graphs of plane bipartite graphs as daisy cubes. https://doi.org/10.1016/j.dam.2025.01.017

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