Flag Bicolorings, Pseudo-Orientations, and Double Covers of Maps

Koike, Hideaki; Pellicer, Daniel; Raggi, Miguel; Wilson, Steve

doi:10.37236/6118

“…In general, if X covers 2 n I it means that there is a coloring of the vertices (or flags) of X with two colors such that i-adjacent flags are of the same color if and only if i ∈ I. See [16,23] for a detailed discussion on flag colorings. If a maniplex X does not cover 2 n I , then X ♦2 n I is a maniplex that covers X but also covers 2 n I .…”

Section: Voltage Operationsmentioning

confidence: 99%

Voltage operations on maniplexes

Hubard¹,

Mochán²,

Montero³

2022

Preprint

0

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Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations on maniplexes, as a way to generalize such operations. Our technique provides a way to study classical operations in a graph theory setting. In fact, voltage operations can be applied to symmetry type graphs, and more generally to n-valent properly n-edge colored graphs. We focus on studying the interactions between voltage operations and the symmetries of the operated object, and show that they can be potentially used to build maniplexes with prescribed symmetry type graphs. Moreover, a complete characterization of when an operation can be seen as a voltage operation is given.

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“…, n − 1}, a bi-colouring of M consistent with I is a colouring of F(M) with colours black and white such that i-adjacent flags have the same colour if and only if i ∈ I. In the context of maps by denoting J := {0, 1, 2} \ I, such a bi-colouring is called an J -colouring in [11].…”

Section: Bi-colourings Consistent With Imentioning

confidence: 99%

An existence result on two-orbit maniplexes

Pellicer

¹

,

Potočnik

²

,

Toledo

³

2019

Journal of Combinatorial Theory, Series A

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A maniplex of rank n is a connected, n-valent, edge-coloured graph that generalises abstract polytopes and maps. If the automorphism group of a maniplex M partitions the vertex-set of M into k distinct orbits, we say that M is a k-orbit n-maniplex. The symmetry type graph of M is the quotient pregraph obtained by contracting every orbit into a single vertex. Symmetry type graphs of maniplexes satisfy a series of very specific properties. The question arises whether any pregraph of order k satisfying these properties is the symmetry type graph of some k-orbit maniplex. We answer the question when k = 2. 1 arXiv:1812.04148v1 [math.CO]

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Voltage Operations on Maniplexes, Polytopes and Maps

Hubard

¹

,

Mochán

²

,

Montero

³

2023

Combinatorica

0

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Classical geometric and topological operations on polyhedra, maps and polytopes often give rise to structures with the same symmetry group as the original one, but with more flags. In this paper we introduce the notion of voltage operations on maniplexes, as a way to unify the study of such operations and generalize them to other geometric and combinatorial structures such as abstract polytopes, hypermaps, maniplexes or hypertopes. This can be done since our technique provides a way to study classical operations in a graph theoretic setting, and thus to apply a voltage operation one only needs that the combinatorial structure in hand can be understood as an n-valent properly n-edge colored graph. For example, in the case of abstract polytopes, the partial order can be encoded into the so-called flag graph of the polytope and the voltage operation is therefore applied to such flag graph to then be recovered as a partial order. We focus on studying the interactions between voltage operations and the symmetries of the operated object, and show that these operations can be potentially used to build maniplexes with prescribed symmetry type graphs. Moreover, a complete characterization of when an operation can be seen as a voltage operation is given.

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Flag Bicolorings, Pseudo-Orientations, and Double Covers of Maps

Abstract: This paper discusses consistent flag bicolorings of maps and maniplexes, in their own right and as generalizations of orientations and pseudo-orientations. Furthermore, a related doubling concept is introduced, and relationships between these ideas are explored. arXiv:1301.4421v1 [math.CO]

Cited by 5 publications

References 9 publications

Voltage operations on maniplexes

Voltage operations on maniplexes

An existence result on two-orbit maniplexes

Voltage Operations on Maniplexes, Polytopes and Maps

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