On the chromatic number of some flip graphs

Fabila-Monroy, Ruy; Flores-Peñaloza, David; Huemer, Clemens; Hurtado, Ferrán; Urrutia, Jorge; Wood, David R.

doi:10.46298/dmtcs.460

Cited by 8 publications

(15 citation statements)

References 14 publications

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“…This means each of the remaining 3m + 4 − y blue diagonals is flipped at least twice. Hence, t III ≥ y + 2(3m + 4 − y) = 6m + 8 − y Together with (3), this becomes (10) y > 1.99m…”

Section: Strong Convexity Fails With Two Flat Verticesmentioning

confidence: 80%

“…In this case, F(Σ) is the graph of a convex polytope-the associahedronwhose dimension is the number of vertices of Σ minus three [17]. The genus of this graph [21], its chromatic number [10], and hamiltonicity [15] have been investigated. Its geometry is specially interesting because of its relation with the rotation distance of binary trees [32,31].…”

Section: Introductionmentioning

confidence: 99%

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Strong convexity in flip-graphs

Pournin¹,

Wang²

2021

Preprint

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A graph structure F (Σ) can be given to the triangulations of a surface Σ with a prescribed set of vertices. Its edges connect two triangulations that differ by a single arc. An interesting question is whether the subgraph Fε(Σ) induced in F (Σ) by the triangulations that contain a given arc ε is strongly convex in the sense that all the geodesic paths between two such triangulations remain in that subgraph. A positive answer to this question has been given when Σ is a convex polygon or a topological surface. Here, we provide a related result that involves a triangle instead of an arc, in the case when Σ is a convex polygon. We show that, when the three edges of a triangle τ appear in (possibly distinct) triangulations along a geodesic path, τ must belong to a triangulation in that path. We also provide two consequences of this result. The first consequence is that Fε(Σ) is not always strongly convex when Σ has either two flat vertices or two punctures. The second is that the number of arc crossings between two triangulations of a convex polygon Σ does not allow to approximate their distance in F (Σ) by a factor of less than 1.25.

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“…This means each of the remaining 3m + 4 − y blue diagonals is flipped at least twice. Hence, t III ≥ y + 2(3m + 4 − y) = 6m + 8 − y Together with (3), this becomes (10) y > 1.99m…”

Section: Strong Convexity Fails With Two Flat Verticesmentioning

confidence: 80%

Section: Introductionmentioning

confidence: 99%

Strong convexity in flip-graphs

Pournin¹,

Wang²

2021

Preprint

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“…We use χ(G) to denote the chromatic number of a graph G. Fabila-Monroy, Flores-Penaloza, Huemer, Hurtado, Urrutia and Wood [9] observed that χ(M(K n,n )) = 2 and using this result, proved that…”

Section: Introductionmentioning

confidence: 99%

On the flip graphs on perfect matchings of complete graphs and signed reversal graphs

Cioabă¹,

Royle²,

Tan³

2020

Preprint

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“…Over recent decades there has been an increasing interest in graphs on combinatorial objects in which the adjacency relation reflects a local change, for example, flip graphs (see [2,4,10,11,13]). In this paper we introduce Yoke graphs, a family of flip graphs that generalizes previously studied families of graphs: colored triangle free triangulations [1] (CTFT), arc permutations [3] and geometric caterpillars [8].…”

Section: Introductionmentioning

confidence: 99%