Geodesic
Improved bounds for some facially constrained colorings
Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 151-158 Cet article a éte moissonné depuis la source Library of Science

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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 2-connected 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_𝓁 )-WORM coloring of a plane graph G is a vertex-coloring such that G contains neither rainbow facial k-path nor monochromatic facial 𝓁-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.
Keywords: plane graph, facial coloring, facial-parity edge-coloring, facial-parity vertex-coloring, WORM coloring 
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Štorgel, Kenny. Improved bounds for some facially constrained colorings. Discussiones Mathematicae. Graph Theory, Tome 43 (2023) no. 1, pp. 151-158. http://geodesic.mathdoc.fr/item/DMGT_2023_43_1_a8/

[1] Cs. Bujtás, E. Sampathkumar, Zs. Tuza, C. Dominic and L. Pushpalatha, Vertex coloring without large polychromatic stars, Discrete Math. 312 (2012) 2102–2108. https://doi.org/10.1016/j.disc.2011.04.013

[2] Cs. Bujtás, E. Sampathkumar, Zs. Tuza, M. Subramanya and C. Dominic, 3-consecutive c-colorings of graphs, Discuss. Math. Graph Theory 30 (2010) 393–405. https://doi.org/10.7151/dmgt.1502

[3] Cs. Bujtás and Zs. Tuza, F-WORM colorings: Results for 2-connected graphs, Discrete Appl. Math. 231 (2017) 131–138. https://doi.org/10.1016/j.dam.2017.05.008

[4] J. Czap, Parity vertex coloring of outerplane graphs, Discrete Math. 311 (2011) 2570–2573. https://doi.org/10.1016/j.disc.2011.06.009

[5] J. Czap, Facial parity edge coloring of outerplane graphs, Ars Math. Contemp. 5 (2012) 289–293. https://doi.org/10.26493/1855-3974.228.ee8

[6] J. Czap, I. Fabrici and S. Jendroľ, Colorings of plane graphs without long monochromatic facial paths, Discuss. Math. Graph Theory 41 (2021) 801–808. https://doi.org/10.7151/dmgt.2319

[7] J. Czap and S. Jendroľ, Facially-constrained colorings of plane graphs: A survey, Discrete Math. 340 (2017) 2691–2703. https://doi.org/10.1016/j.disc.2016.07.026

[8] J. Czap, S. Jendroľ and F. Kardoš, Facial parity edge colouring, Ars Math. Contemp. 4 (2011) 255–269. https://doi.org/10.26493/1855-3974.129.be3

[9] J. Czap, S. Jendroľ, F. Kardoš and R. Soták, Facial parity edge colouring of plane pseudographs, Discrete Math. 312 (2012) 2735–2740. https://doi.org/10.1016/j.disc.2012.03.036

[10] J. Czap, S. Jendroľ and J. Valiska, WORM colorings of planar graphs, Discuss. Math. Graph Theory 37 (2017) 353–368. https://doi.org/10.7151/dmgt.1921

[11] J. Czap, S. Jendroľ and M. Voigt, Parity vertex colouring of plane graphs, Discrete Math. 311 (2011) 512–520. https://doi.org/10.1016/j.disc.2010.12.008

[12] W. Goddard, K. Wash and H. Xu, WORM colorings forbidding cycles or cliques, Congr. Numer. 219 (2014) 161–173.

[13] W. Goddard, K. Wash and H. Xu, WORM colorings, Discuss. Math. Graph Theory 35 (2015) 571–584. https://doi.org/10.7151/dmgt.1814

[14] T. Kaiser, O. Rucký, M. Stehlík and R. Škrekovski, Strong parity vertex coloring of plane graphs, Discrete Math. Theor. Comput. Sci. 16 (2014) 143–158.

[15] B. Lužar and R. Škrekovski, Improved bound on facial parity edge coloring, Discrete Math. 313 (2013) 2218–2222. https://doi.org/10.1016/j.disc.2013.05.022

[16] Zs. Tuza, Graph colorings with local constraints–-A survey, Discuss. Math. Graph Theory 17 (1997) 161–228. https://doi.org/10.7151/dmgt.1049

[17] V. Voloshin, The mixed hypergraphs, Comput. Sci. J. Moldova 1 (1993) 45–52.

[18] W. Wang, S. Finbow and P. Wang, An improved bound on parity vertex colourings of outerplane graphs, Discrete Math. 312 (2012) 2782–2787. https://doi.org/10.1016/j.disc.2012.04.009