Resonance graphs of plane bipartite graphs as daisy cubes

TL;DR

This work characterizes when resonance graphs of plane bipartite graphs are daisy cubes. It proves that for plane elementary bipartite graphs $G\neq K_2$, the resonance graph $R(G)$ is a daisy cube if and only if the Fries number $Fr(G)$ equals the number of finite faces of $G$, equivalently $G$ is peripherally $2$-colorable with that many finite faces. The results extend to general plane bipartite graphs by requiring weakly elementary structure and that each non-$K_2$ elementary component $G_i$ satisfies $Fr(G_i)$ equal to its finite-face count; in this case $R(G)$ factors as the Cartesian product of the $R(G_i)$, each a daisy cube. Overall, the paper links combinatorial resonance properties to daisy-cube structure and clarifies how Cartesian products preserve the daisy-cube property.

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.

Figures (5)

• Figure 1: A peripherally 2-colorable graph that is 2-connected outerplane bipartite.
• Figure 2: The resonance graph $R(G)$ of a kinky benzenoid graph $G$.
• Figure 3: A peripherally 2-colorable graph $G$ is transformed into a 2-connected outerplane bipartite graph $G'$ that is also peripherally 2-colorable by edge subdivisions and smoothings.
• Figure 4: A plane bipartite graph $G$ with two elementary components $G_1$ and $G_2$.
• Figure 5: Resonance graphs of graphs $G_1$, $G_2$, and $G$ from Figure \ref{['sl5']}.

Theorems & Definitions (14)

• Lemma 2.1
• Lemma 2.2
• Theorem 2.3
• Theorem 2.4
• Lemma 3.1
• Definition 3.2
• Lemma 3.3
• Lemma 3.4
• Theorem 3.5
• Corollary 3.6