I was looking at coloring a representation of a snark graph as a drawing with crossings. I colored the arcs with two rules: if two arcs meet at the same vertex they have a distinct color. If three arcs meet at a crossing they are all the same color or distinct.

Here is a coloring of a particular drawings of the Petersen graph.


snark coloring

But I am curious, has this coloring of graph drawings been studied?


Snarks are connected, bridgeless cubic graphs which can't be edge colored in three colors. One reason they are important is because proving any instance of the cycle double cover conjecture reduces to solving it for a snark.

My coloring is related to the Fox 3-coloring of knots. https://en.m.wikipedia.org/wiki/Fox_n-coloring It's for knots (which have no vertices)

As with knots at most two edges cross at any point. I should have explicitly stated this.

  • $\begingroup$ A general CONDITION: the number of different colors of $n$ arcs at a cross-section has the same parity as $n$. $\endgroup$ – Włodzimierz Holsztyński Sep 6 '16 at 5:32
  • $\begingroup$ What is a snark? $\endgroup$ – Nick Gill Sep 6 '16 at 7:46
  • $\begingroup$ @Nick Normally a snark is a cyclically 4-edge connected cubic graph with girth at least 5 and chromatic index 4. $\endgroup$ – Gordon Royle Sep 6 '16 at 7:50
  • 1
    $\begingroup$ I'm guessing that your invariant depends on the particular projection, and is not an isotopy invariant of a snark embedding into R^3. 3-colorings correspond to reps. to the dihedral group of order 6, whereas the sort of vertex colorings of cubic graphs seem to relate to colorings by the non-trivial elements of the Klein 4 group. In any case, it appears that your colorings don't have local modifications when graph Reidemeister moves are performed (an edge passes under a vertex). $\endgroup$ – Ian Agol Sep 6 '16 at 17:08

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