Improved bound on facial parity edge coloring

Lužar, Borut; Škrekovski, Riste

doi:10.1016/j.disc.2013.05.022

Cited by 10 publications

(6 citation statements)

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“…A generalization of Theorem 1.1 was successfully applied in solving a problem of facial parity edge colorings in [2], and its improvement in [3]. In this paper, an analogous result to Theorem 1.1 is proved for loopless graphs.…”

Section: Introductionmentioning

confidence: 71%

“…In [5], Pyber proved that 4 colors suffice for an odd edge coloring of any simple graph. Recently, some results on this type of colorings of (multi)graphs were successfully applied in solving a problem of facial parity edge coloring [3,2]. In this paper we present additional results, namely we prove that 6 colors suffice for an odd edge coloring of any loopless connected (multi)graph, provide examples showing that this upper bound is sharp and characterize the family of loopless connected (multi)graphs for which the bound 6 is achieved.…”

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confidence: 87%

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Odd edge coloring of graphs

2015

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An edge coloring of a graph G is said to be an odd edge coloring if for each vertex v of G and each color c, the vertex v uses the color c an odd number of times or does not use it at all. In [5], Pyber proved that 4 colors suffice for an odd edge coloring of any simple graph. Recently, some results on this type of colorings of (multi)graphs were successfully applied in solving a problem of facial parity edge coloring [3,2]. In this paper we present additional results, namely we prove that 6 colors suffice for an odd edge coloring of any loopless connected (multi)graph, provide examples showing that this upper bound is sharp and characterize the family of loopless connected (multi)graphs for which the bound 6 is achieved. We also pose several open problems.

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Section: Introductionmentioning

confidence: 71%

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confidence: 87%

Odd edge coloring of graphs

2015

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“…This bound was later improved by Czap et al [9] to 20 colors. The best known upper bound so far is 16 colors, due to Lužar andŠkrekovski [15]. In [5], an example of an outerplane graph is presented, namely two cycles C 5 sharing a single vertex, which needs 10 colors.…”

Section: Facial-parity Edge-coloringmentioning

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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“…An odd facial edge-coloring of a plane graph is a facial edge-coloring such that for every face f and every color c, either no edge or an odd number of edges incident with f is colored by c. Czap et al [4] proved that every 2-edge-connected plane graph G has an odd facial edge-coloring with at most 20 colors, this bound was later improved to 16 by Lužar and Škrekovski [11]. In the case when G is a 3-edge-connected (resp.…”

Section: Introduction and Notationsmentioning

confidence: 99%

Odd facial colorings of acyclic plane graphs

Czap

Šugerek

2021

EJGTA

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Let G be a connected plane graph with vertex set V and edge set E. For X ∈ {V, E, V ∪ E}, two elements of X are facially adjacent in G if they are incident elements, adjacent vertices, or facially adjacent edges (edges that are consecutive on the boundary walk of a face of G). A coloring of G is facial with respect to X if there is a coloring of elements of X such that facially adjacent elements of X receive different colors. A facial coloring of G is odd if for every face f and every color c, either no element or an odd number of elements incident with f is colored by c. In this paper we investigate odd facial colorings of trees. The main results of this paper are the following: (i) Every tree admits an odd facial vertex-coloring with at most 4 colors; (ii) Only one tree needs 6 colors, the other trees admit an odd facial edge-coloring with at most 5 colors; and (iii) Every tree admits an odd facial total-coloring with at most 5 colors. Moreover, all these bounds are tight.

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Improved bound on facial parity edge coloring

Cited by 10 publications

References 5 publications

Odd edge coloring of graphs

Odd edge coloring of graphs

Improved bounds for some facially constrained colorings

Odd facial colorings of acyclic plane graphs

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