Geometric bijections between spanning subgraphs and orientations of a graph

Changxin Ding

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Geometric bijections between spanning subgraphs and orientations of a graph

Changxin Ding

Combinatorics · Submitted 4 Sep 2021

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim In this paper, we extend the geometric bijections to subgraph-orientation correspondences.

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arXiv listing abstract v4. Results and the main claim are underlined; show every role.

Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called \emph{geometric bijections}. In this paper, we extend the geometric bijections to subgraph-orientation correspondences. Moreover, we extend the geometric constructions accordingly. Our proofs are purely combinatorial, even for the geometric constructions. We also provide geometric proofs for partial results, which make use of zonotopal tiling, relate to Backman, Baker, and Yuen's method, and motivate our combinatorial constructions. Finally, we explain that the main results hold for \emph{regular matroids}.

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What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Let $G$ be a connected finite graph.	background 1.00	0.08	0.01	asserted 0.84
s1	Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^}$ between spanning trees of $G$ and $(\sigma,\sigma^)$-compatible orientations, where the $(\sigma,\sigma^)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^$.	background 1.00	0.03	0.06	asserted 1.00
s2	Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called \emph{geometric bijections}.	background 0.96	0.07	0.03	asserted 1.00
s3	In this paper, we extend the geometric bijections to subgraph-orientation correspondences.	contribution 1.00	0.94	0.42	asserted 1.00
s4	Moreover, we extend the geometric constructions accordingly.	contribution 0.94	0.93	0.09	asserted 0.99
s5	Our proofs are purely combinatorial, even for the geometric constructions.	method 0.88	0.93	0.08	asserted 1.00
s6	We also provide geometric proofs for partial results, which make use of zonotopal tiling, relate to Backman, Baker, and Yuen's method, and motivate our combinatorial constructions.	method 0.69	0.94	0.06	asserted 1.00
s7	Finally, we explain that the main results hold for \emph{regular matroids}.	result 0.64	0.86	0.34	asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.98 · evidence from proof or derivation 1.00.

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The paper

Original abstract

Let $G$ be a connected finite graph. Backman, Baker, and Yuen have constructed a family of explicit and easy-to-describe bijections $g_{\sigma,\sigma^*}$ between spanning trees of $G$ and $(\sigma,\sigma^*)$-compatible orientations, where the $(\sigma,\sigma^*)$-compatible orientations are the representatives of equivalence classes of orientations up to cycle-cocycle reversal which are determined by a cycle signature $\sigma$ and a cocycle signature $\sigma^*$. Their proof makes use of zonotopal subdivisions and the bijections $g_{\sigma,\sigma^*}$ are called \emph{geometric bijections}. In this paper, we extend the geometric bijections to subgraph-orientation correspondences. Moreover, we extend the geometric constructions accordingly. Our proofs are purely combinatorial, even for the geometric constructions. We also provide geometric proofs for partial results, which make use of zonotopal tiling, relate to Backman, Baker, and Yuen's method, and motivate our combinatorial constructions. Finally, we explain that the main results hold for \emph{regular matroids}.

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1 arXiv paper cite this in 20 sentences

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Papers citing it · 13 sentences without a confident label

Builds on it 1 paper

By the two theorems and a bit extra work, we have the following theorems, which generalize the work in [ref] and [ref] , respectively.

A framework unifying some bijections for graphs and its connection to Lawrence polytopes · Signatures and the two atlases · 1.00

Compares with it 1 paper

To reformulate Theorem [ref] and Theorem [ref] in terms of signatures and compatible orientations, we write [equation] They are exactly the BBY bijection in [ref] and the extened BBY bijection in [ref] when the two signatures are acyclic.

A framework unifying some bijections for graphs and its connection to Lawrence polytopes · Signatures and the two atlases · 0.90

Cites it as background 1 paper

First we need to recall some notions and results in [ref] . not the same cube Consider the hypercube [equation] and the half-open cell [equation] where $O$ is an orientation and $A⊆E$.

A framework unifying some bijections for graphs and its connection to Lawrence polytopes · A tiling induced by the main bijection $φ_𝒜,𝒜^*$ · 0.95

What it cites · 2 sentences without a confident label

Cites as background 1 paper

For the space tiling zonotopes, see [ref] ; for the decomposition of the zonotope into half-open cells, see [ref] .

Once more about Voronoi's conjecture and space tiling zonotopes · Introduction to the geometric ideas behind the main theorem with an example · 0.93

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