Large planar $(n,m)$-cliques

TL;DR

This paper establishes a tight upper bound on the number of vertices in planar $(n,m)$-complete graphs, confirming a conjecture and advancing understanding of labeled graph structures in planarity constraints.

Contribution

It proves a maximum vertex bound for planar $(n,m)$-complete graphs, settling a conjecture and providing a precise formula for all cases except $(0,1)$.

Findings

Maximum of $3(2n+m)^2+(2n+m)+1$ vertices for planar $(n,m)$-complete graphs

Bound is tight for all $(n,m) eq (0,1)$

Confirmed a conjecture by Bensmail et al. (2017)

Abstract

An \textit{$(n,m)$-graph} $G$ is a graph having both arcs and edges, and its arcs (resp., edges) are labeled using one of the $n$ (resp., $m$) different symbols. An \textit{$(n,m)$-complete graph} $G$ is an $(n,m)$-graph without loops or multiple edges in its underlying graph such that identifying any pair of vertices results in a loop or parallel adjacencies with distinct labels. We show that a planar $(n,m)$-complete graph cannot have more than $3(2n+m)^2+(2n+m)+1$ vertices, for all $(n,m) \neq (0,1)$ and that the bound is tight. This positively settles a conjecture by Bensmail \textit{et al.}~[Graphs and Combinatorics 2017].