Facially-constrained colorings of plane graphs: A survey

Czap, Július; Jendroľ, Stanislav

doi:10.1016/j.disc.2016.07.026

Cited by 27 publications

(17 citation statements)

References 110 publications

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“…Fabrici and Göring used this concept in a facial version, which is of great interest, among others, also due to Conjecture 1 and its direct connection to the Four Color Theorem. Coloring embedded graphs with respect to faces is a bursting field itself; the main directions are presented in a recent survey by Czap and Jendrol' [4].…”

Section: Introductionmentioning

confidence: 99%

On facial unique-maximum (edge-)coloring

Andova

Lidický

Lužar

et al. 2018

Discrete Applied Mathematics

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A facial unique-maximum coloring of a plane graph is a vertex coloring where on each face α the maximal color appears exactly once on the vertices of α. If the coloring is required to be proper, then the upper bound for the minimal number of colors required for such a coloring is set to 5. Fabrici and Göring [5] even conjectured that 4 colors always suffice. Confirming the conjecture would hence give a considerable strengthening of the Four Color Theorem. In this paper, we prove that the conjecture holds for subcubic plane graphs, outerplane graphs and plane quadrangulations. Additionally, we consider the facial edge-coloring analogue of the aforementioned coloring and prove that every 2-connected plane graph admits such a coloring with at most 4 colors.

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Section: Introductionmentioning

confidence: 99%

On facial unique-maximum (edge-)coloring

Andova

Lidický

Lužar

et al. 2018

Discrete Applied Mathematics

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“…In [5], an example of an outerplane graph is presented, namely two cycles C 5 sharing a single vertex, which needs 10 colors. Later Czap and Jendroľ [7] proposed the following conjecture.…”

Section: Facial-parity Edge-coloringmentioning

confidence: 96%

“…Among some 2 K.Štorgel of the recent studies are those that study colorings of plane graphs where certain constraints are given by the faces. Czap and Jendroľ [7] wrote a survey devoted to presenting many of such colorings, in which they also presented a number of open problems. In this note, we consider three types of facially constrained colorings of plane graphs.…”

Section: Introductionmentioning

confidence: 99%

Improved bounds for some facially constrained colorings

Štorgel¹

2020

Discuss. Math. Graph Theory

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A facial-parity edge-coloring of a 2-edge-connected plane graph is a facially-proper edge-coloring in which every face is incident with zero or an odd number of edges of each color. A facial-parity vertex-coloring of a 2connected plane graph is a proper vertex-coloring in which every face is incident with zero or an odd number of vertices of each color. Czap and Jendroľ in [Facially-constrained colorings of plane graphs: A survey, Discrete Math. 340 (2017) 2691-2703], conjectured that 10 colors suffice in both colorings. We present an infinite family of counterexamples to both conjectures. A facial (P k , P)-WORM coloring of a plane graph G is a vertex-coloring such that G contains neither rainbow facial k-path nor monochromatic facial-path. Czap, Jendroľ and Valiska in [WORM colorings of planar graphs, Discuss. Math. Graph Theory 37 (2017) 353-368], proved that for any integer n ≥ 12 there exists a connected plane graph on n vertices, with maximum degree at least 6, having no facial (P 3 , P 3)-WORM coloring. They also asked whether there exists a graph with maximum degree 4 having the same property. We prove that for any integer n ≥ 18, there exists a connected plane graph, with maximum degree 4, with no facial (P 3 , P 3)-WORM coloring.

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“…Two edges are facially adjacent in G if they are consecutive edges on a boundary walk of a face of G. Colorings of graphs embedded in the plane with face-constrains have recently drawn a substantial amount of attention, see e.g. [4]. A facial edge-coloring of a plane graph G is a mapping ϕ : E(G) → {1, .…”

Section: Introductionmentioning

confidence: 99%

Facial Total-Coloring of Bipartite Plane Graphs

Czap¹,

Šugerek²

2017

Int. J. of Pure and Appl. Math.

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Abstract:Let G be a plane graph. Two edges are facially adjacent in G if they are consecutive edges on a boundary walk of a face of G. A facial edge-coloring of G is an edge-coloring such that any two facially adjacent edges receive distinct colors. A facial totalcoloring of G is a coloring of vertices and edges such that no facially adjacent edges, no adjacent vertices, and no edge and its endvertices are assigned the same color. In this paper we prove that every plane graph admits a facial edge-coloring with at most four colors such that at most three colors appear at each vertex. Using this result we confirm a conjecture on facial total-coloring for bipartite plane graphs.

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Facially-constrained colorings of plane graphs: A survey

Cited by 27 publications

References 110 publications

On facial unique-maximum (edge-)coloring

On facial unique-maximum (edge-)coloring

Improved bounds for some facially constrained colorings

Facial Total-Coloring of Bipartite Plane Graphs

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