### Odd facial colorings of acyclic plane graphs

#### Abstract

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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PDFDOI: http://dx.doi.org/10.5614/ejgta.2021.9.2.8

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