Geodesic
Facial [r,s,t]-Colorings of Plane Graphs
Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 3, pp. 629-645

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Let G be a plane graph. Two edges are facially adjacent in G if they are consecutive edges on the boundary walk of a face of G. Given nonnegative integers r, s, and t, a facial [r, s, t]-coloring of a plane graph G = (V,E) is a mapping f : V ∪ E →1, . . ., k such that |f(v_1) − f(v_2)| ≥ r for every two adjacent vertices v_1 and v_2, | f(e_1) − f(e_2)| ≥ s for every two facially adjacent edges e_1 and e_2, and | f(v) − f(e)| ≥ t for all pairs of incident vertices v and edges e. The facial [r, s, t]-chromatic number χ_r,s,t (G) of G is defined to be the minimum k such that G admits a facial [r, s, t]-coloring with colors 1, . . ., k. In this paper we show that χ_r,s,t (G) ≤ 3r + 3s + t + 1 for every plane graph G. For some triplets [r, s, t] and for some families of plane graphs this bound is improved. Special attention is devoted to the cases when the parameters r, s, and t are small.
Keywords: plane graph, boundary walk, edge-coloring, vertex-coloring, total-coloring 
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     author = {Czap, J\'ulius and \v{S}ugerek, Peter and Jendrol{\textquoteright}, Stanislav and Valiska, Juraj},
     title = {Facial {[r,s,t]-Colorings} of {Plane} {Graphs}},
     journal = {Discussiones Mathematicae. Graph Theory},
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     publisher = {mathdoc},
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     year = {2019},
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Czap, Július; Šugerek, Peter; Jendrol’, Stanislav; Valiska, Juraj. Facial [r,s,t]-Colorings of Plane Graphs. Discussiones Mathematicae. Graph Theory, Tome 39 (2019) no. 3, pp. 629-645. http://geodesic.mathdoc.fr/item/DMGT_2019_39_3_a1/