Odd coloring of 2-boundary planar graphs and beyond

Abstract

In this paper, we introduce the notion of 2-boundary planar graphs. A graph is 2-boundary planar if it has an embedding in the plane so that all vertices lie on the boundary of at most two faces and no edges are crossed. A proper coloring of a graph is odd if every non-isolated vertex has some color that appears an odd number of times on its neighborhood. Petruševski and Škrekovski conjectured in 2022 that every planar graph admits an odd 5-coloring. We confirm this conjecture for 2-boundary planar graphs. Moreover, we present several questions regarding 2-boundary planar graphs that are of independent interest.

18 pages, 5 figures. Reorganize the paper to place greater emphasis on the newly introduced 2-boundary planar graphs; incorporate various problems, questions, or conjectures related to 2-boundary planar graphs to facilitate further research; and include an additional author