Bounding Clique Size in Squares of Planar Graphs

TL;DR

This paper establishes an upper bound on the clique size in the square of planar graphs with high maximum degree, advancing understanding of graph coloring and clique bounds in planar graph theory.

Contribution

It proves a new bound on the clique number of the square of planar graphs with maximum degree at least 36, based on a structural lemma involving three vertices.

Findings

Bound on  in (G^2) for planar graphs with   36

Existence of three vertices characterizing large maximal cliques in G^2

Structural lemma linking clique size to neighborhood intersections

Abstract

Wegner conjectured that if $G$ is a planar graph with maximum degree $\Delta\ge 8$, then $\chi(G^2)\le \left\lfloor \frac32\Delta\right\rfloor +1$. This problem has received much attention, but remains open for all $\Delta\ge 8$. Here we prove an analogous bound on $\omega(G^2)$: If $G$ is a plane graph with $\Delta(G)\ge 36$, then $\omega(G^2)\le \lfloor\frac32\Delta(G)\rfloor+1$. In fact, this is a corollary of the following lemma, which is our main result. If $G$ is a plane graph with $\Delta(G)\ge 19$ and $S$ is a maximal clique in $G^2$ with $|S|\ge \Delta(G)+20$, then there exist $x,y,z\in V(G)$ such that $S=\{w:|N[w]\cap\{x,y,z\}|\ge 2\}$.