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All Journal Electronic Journal of Graph Theory and Applications (EJGTA)
Peter Šugerek
Department of Applied Mathematics and Business Informatics, Faculty of Economics, Technical University of Košice, Němcovej 32, 04001 Košice, Slovakia
Electrical & Electronics Engineering

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Odd facial colorings of acyclic plane graphs Július Czap; Peter Šugerek
Electronic Journal of Graph Theory and Applications (EJGTA) Vol 9, No 2 (2021): Electronic Journal of Graph Theory and Applications
Publisher : GTA Research Group, Univ. Newcastle, Indonesian Combinatorics Society and ITB

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.5614/ejgta.2021.9.2.8

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.
Co-Authors Július Czap