Let G be a plane bipartite graph admitting a perfect matching, and let R(G) denote its resonance graph. It is known that any connected resonance graph can be isometrically embedded into hypercubes as a finite distributive lattice. Let <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n_0(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> denote the number of finite faces of G without forbidden edges on their peripheries. We show that any connected R(G) has isometric dimension at least <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n_0(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>, and the lower bound is attained if and only if G is a plane weakly elementary bipartite graph such that the infinite face of each nontrivial elementary component of G is forcing. We also design an algorithm to produce a binary coding on the vertex set of R(G) which induces an isometric embedding of R(G) as a finite distributive lattice into a hypercube of dimension <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n_0(G)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mn>0</mn> </msub> <mrow> <mo stretchy="false">(</mo> <mi>G</mi> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>.

Isometric embeddings of resonance graphs as finite distributive lattices

期刊论文
出版模式：

开放获取
发表日期：

2026年7月3日

查看全文

Zhongyuan Che

摘要

Let G be a plane bipartite graph admitting a perfect matching, and let R(G) denote its resonance graph. It is known that any connected resonance graph can be isometrically embedded into hypercubes as a finite distributive lattice. Let $n_0(G)$ $n_{0} (G)$ denote the number of finite faces of G without forbidden edges on their peripheries. We show that any connected R(G) has isometric dimension at least $n_0(G)$ $n_{0} (G)$ , and the lower bound is attained if and only if G is a plane weakly elementary bipartite graph such that the infinite face of each nontrivial elementary component of G is forcing. We also design an algorithm to produce a binary coding on the vertex set of R(G) which induces an isometric embedding of R(G) as a finite distributive lattice into a hypercube of dimension $n_0(G)$ $n_{0} (G)$ .