Odd coloring of 2-boundary planar graphs and beyond
Weichan Liu, Mengke Qi, Xin Zhang
TL;DR
This work introduces 2-boundary planar graphs and proves that they all admit an odd $5$-coloring, thereby confirming the conjecture of Petruševski and Škrekovski for this graph class. The authors employ a structural-reduction approach, defining embedding features such as $f_{out}$, $f_{in}$, shared vertices, and the parameter $\tilde{\Delta}(G)$, and identifying a finite set of odd $k$-reducible configurations to rule out potential minimal counterexamples. They establish the main theorem via exhaustive reduction of these configurations for $k\ge 5$ and study the weak dual of 2-boundary plane graphs, showing it is a forest or unicyclic, with implications for related graph classes and minor-closed questions. The results connect to edge-coloring theory (e.g., bounds on chromatic index for certain subgraphs) and raise open problems on minor-characterizations and linear-time recognition/embedding of 2-boundary planar graphs.
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.