The Extremal Function and Colin de Verdière Graph Parameter

  • Rose McCarty
Keywords: Graph theory, Planar graphs, Graph parameters, Colin de Verdière parameter, Extremal graph theory, Extremal function, Graph complements, Chordal graphs, Graph minors


The Colin de Verdière parameter $\mu(G)$ is a minor-monotone graph parameter with connections to differential geometry. We study the conjecture that for every integer $t$, if $G$ is a graph with at least $t$ vertices and $\mu(G) \leq t$, then $|E(G)| \leq t|V(G)|-\binom{t+1}{2}$. We observe a relation to the graph complement conjecture for the Colin de Verdière parameter and prove the conjectured edge upper bound for graphs $G$ such that either $\mu(G) \leq 7$, or $\mu(G) \geq |V(G)|-6$, or the complement of $G$ is chordal, or $G$ is chordal.

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