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Bijective proofs of proper coloring theorems

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Bruce E. Sagan (Michigan State University), Vincent Vatter (University of Florida)

Combinatorics · Submitted 27 Jul 2020

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

The authors' main claim We establish all six of these expressions bijectively.

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arXiv ID2007.13725 (version 1)
Pith ID376ZVKXV, Pith's permanent record of this paper and its review
CitationBruce E. Sagan (Michigan State University), Vincent Vatter (University of Florida). "Bijective proofs of proper coloring theorems." arXiv:2007.13725 (2020). https://pith.science/paper/376ZVKXV
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What the abstract itself states, as probabilities: claims first 0.06 · states limitation 0.12 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.80 · compares with prior results 0.57 · mentions code or data availability 0.02.

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arXiv listing abstract v1. Results and the main claim are underlined; .

The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.

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#sentencerolethe authors' own workstates the main conclusionstrength
s0 The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. background 1.00 0.03 0.02 asserted 1.00
s1 Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. background 1.00 0.03 0.04 asserted 1.00
s2 We establish all six of these expressions bijectively. contribution 0.85 0.97 0.87 asserted 1.00
s3 In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs. method 0.55 0.95 0.26 asserted 1.00

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

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The paper
Original abstract

The chromatic polynomial and its generalization, the chromatic symmetric function, are two important graph invariants. Celebrated theorems of Birkhoff, Whitney, and Stanley show how both objects can be expressed in three different ways: as sums over all spanning subgraphs, as sums over spanning subgraphs with no broken circuits, and in terms of acyclic orientations with compatible colorings. We establish all six of these expressions bijectively. In fact, we do this with only two bijections, as the proofs in the symmetric function setting are obtained using the same bijections as in the polynomial case and the bijection for broken circuits is just a restriction of the one for all spanning subgraphs.

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