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I've seen some statements that trivalent graphs in a surface are 'generic'. See for example the Wiki entry on cubic graphs.

I'm wondering how this could be rephrased. Here are some (somewhat imprecise) ways I've seen 'generic' be used in the context of intersections of submanifolds.

  • An intersection is generic if in some local coordinates, the intersections are defined by the zero set of some linear equations. For example, the curve $y=x^2$ intersecting the $x$-axis is not generic since there's a double-zero at the intersection point. However, $y=x^2 - a, a>0$ does intersect the $x$-axis generically since the neighborhood around the intersection points looks locally like two lines crossing.
  • A generic intersection as described above can be gotten from an arbitrary intersection by almost-all perturbations of the manifolds. For example, shifting $y=x^2$ up or down slightly either removes the intersection or creates two locally linear intersections.

Can the idea of trivalent graphs being 'generic' be phrased in these terms? More precisely:

  1. Can graphs embedded in a plane be defined by some algebraic equations that only have 'generic' solutions if the graph is trivalent?
  2. Is there a sense in which we can perturb a $d$-valent vertex with $d>3$ so that the perturbed vertex gets split into $(d-1)$ trivalent vertices?
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  • $\begingroup$ For 2, one can consider graphs as closed subsets of the surface and define a metric on the set of graphs using Hausdorff distance. $\endgroup$
    – Will Sawin
    Commented Mar 28, 2021 at 23:50
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    $\begingroup$ math.stackexchange.com/questions/2363710/… $\endgroup$
    – Chris Gerig
    Commented Mar 28, 2021 at 23:54
  • $\begingroup$ Chris, thank you, that's helpful. And @Will, can you elaborate on this? Wouldn't the Hausdorff distance of two graphs that intersect each other be zero? And how would this 'split up' a vertex in the sense I described? $\endgroup$
    – Joe
    Commented Mar 29, 2021 at 0:04
  • $\begingroup$ The Hausdorff distance is defined as the maximum distance from a point in one set to the other set, not the minimum distance, so it only vanishes when the two sets are equal. It is easy to see that every graph is close in the Hausdorff distance to a trivalent graph - just take one of the edges near the vertex, perturb it so it now hits the remaining edges somewhere other than the vertex, and repeat. $\endgroup$
    – Will Sawin
    Commented Mar 29, 2021 at 0:53

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