Edge-coloring of plane multigraphs with many colors on facial cycles

For a fixed positive integer p, a coloring of the edges of a multigraph G is called p-acyclic coloring if every cycle C in G contains at least min{vertical bar C vertical bar, p + 1} colors. The least number of colors needed for a p-acyclic coloring of G is the p-arboricity of G. From a result of Bartnicki et al. (2019) it follows that there are planar graphs with unbounded p-arboricity. In this note we relax the definition of p-arboricity for plane multigraphs in sense that the requirement is not for all cycles but only for facial cycles and show that the smallest number of colors needed for such a coloring is a constant (depending on p only). (C) 2019 Elsevier B.V. All rights reserved.