Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Fernandes, Cristina G.; Pina, José Coelho de; Alfonsín, Jorge Luis Ramírez; Robins, Sinai

doi:10.1007/s00454-020-00192-1

Cited by 3 publications

(10 citation statements)

References 15 publications

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“…A better understanding of the connection between Ehrhart equivalence and rational discrete equidecomposability for various classes of polytopes is a fascinating avenue for future research. In particular, the recent work [4] showed that the two notions are equivalent for integral polytopes in R 3 , and the recent work [5] showed the same is true for polytopes associated to cubic graphs.…”

Section: Resultsmentioning

confidence: 96%

Discrete Equidecomposability and Ehrhart Theory of Polygons

Turner

2020

Discrete Comput Geom

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Motivated by questions from Ehrhart theory, we present new results on discrete equidecomposability. Two rational polygons P and Q are said to be discretely equidecomposable if there exists a piecewise affine-unimodular bijection (equivalently, a piecewise affine-linear bijection that preserves the integer lattice Z 2 ) from P to Q. We develop an invariant for a particular version of this notion called rational finite discrete equidecomposability. We construct triangles that are Ehrhart equivalent but not rationally finitely discretely equidecomposable, thus providing a partial negative answer to a question of Haase-McAllister on whether Ehrhart equivalence implies discrete equidecomposability. Surprisingly, if we delete an edge from each of these triangles, there exists an infinite rational discrete equidecomposability relation between them. Our final section addresses the topic of infinite equidecomposability with concrete examples and a potential setting for further investigation of this phenomenon.

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Section: Resultsmentioning

confidence: 96%

Discrete Equidecomposability and Ehrhart Theory of Polygons

Turner

2020

Discrete Comput Geom

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“…Although our approach allows us to control the behavior for the difference of polynomials, we were not able, despite many efforts, to find a method to compute the desired polynomials explictly. This seems a challenging task even for {1, 3}-trees as stated in [6,Problem 6.5].…”

Section: Hightlighting Context Of Our Main Resultsmentioning

confidence: 99%

“…Note that a, b, c, and d are not necessarily pairwise distinct. The weight function w , defined as in [6], is such that w f = w f for every f = e and…”

Section: Nearest Neighbor Interchangesmentioning

confidence: 99%

“…The composition of the corresponding bijections is a bijection between part t (G h,k , H) and part t (G h,k , H ). Indeed, the used NNIs are as in Corollary 10 from [6], and preserve the parity of the values on the corresponding leaf-edges and loops. So the lemma follows.…”

Section: Lemma 17 If G Is a Connected {1 3}-graph With H Leaf-edges A...mentioning

confidence: 99%

“…They were mainly motivated by the relation of these quasi-polynomials to the study of dormant torally indigenous bundles on a general curve, objects arising in algebraic geometry [14]. This connection was further investigated in [17], and more properties of the polytope P G were presented in [6].…”

Section: Introductionmentioning

confidence: 99%

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On the period collapse of a family of Ehrhart quasi-polynomials

Fernandes,

de Pina,

Alfonsín

et al. 2021

Preprint

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A graph whose nodes have degree 1 or 3 is called a {1, 3}-graph. Liu and Osserman associated a polytope to each {1, 3}-graph and studied the Ehrhart quasipolynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set {0, 1 4 , 1 2 , 1}, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasipolynomial of these polytopes is at most 2 if the graph is a tree or a cubic graph, and it is equal to 4 otherwise.In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasi-polynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.

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Period collapse in Ehrhart quasi-polynomials of \(\{1,3\}\)-graphs

Fernandes¹,

Pina²,

Alfonsín³

et al. 2022

Combinatorial Theory

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A graph whose nodes have degree 1 or 3 is called a {1, 3}-graph. Liu and Osserman associated a polytope to each {1, 3}-graph and studied the Ehrhart quasi-polynomials of these polytopes. They showed that the vertices of these polytopes have coordinates in the set {0, 1 4 , 1 2 , 1}, which implies that the period of their Ehrhart quasi-polynomials is either 1, 2, or 4. We show that the period of the Ehrhart quasi-polynomial of these polytopes is 2 if the graph is a tree, the period is at most 2 if the graph is cubic, and the period is 4 otherwise.In the process of proving this theorem, several interesting combinatorial and geometric properties of these polytopes were uncovered, arising from the structure of their associated graphs. The tools developed here may find other applications in the study of Ehrhart quasipolynomials and enumeration problems for other polytopes that arise from graphs. Additionally, we have identified some interesting connections with triangulations of 3-manifolds.

show abstract

Cubic Graphs, Their Ehrhart Quasi-Polynomials, and a Scissors Congruence Phenomenon

Cited by 3 publications

References 15 publications

Discrete Equidecomposability and Ehrhart Theory of Polygons

Discrete Equidecomposability and Ehrhart Theory of Polygons

On the period collapse of a family of Ehrhart quasi-polynomials

Period collapse in Ehrhart quasi-polynomials of \(\{1,3\}\)-graphs

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