Your article is very nicely done. I love the observation that double 3's acts as an identity. I have a few questions if you don't mind.

Firstly, you present a very helpful multiplication table:

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`000 0 100 0 200 0 300 0`

010 0 110 0 210 0 310 0

020 0 120 0 220 0 320 0

030 0 130 0 230 0 330 0

001 0 101 0 201 0 301 0

011 0 111 0 211 0 311 0

021 0 121 1 221 0 321 1

031 0 131 1 231 0 331 1

002 0 102 0 202 0 302 0

012 0 112 0 212 2 312 2

022 0 122 0 222 0 322 0

032 0 132 0 232 2 332 2

003 0 103 0 203 0 303 0

013 0 113 0 213 2 313 2

023 0 123 1 223 0 323 1

033 0 133 1 233 2 333 3

and write, for example that

gsf wrote:...a segment { 1-edge 3-edge 1-edge } (denoted 131) is equivalent to a single 1-edge.

In the case where the segment is even a tri-cycle, one obtains a 1-edge from the "exceptional" vertex to itself, implying that the vertex is not the induced subgraph color. Is this correct? The reason I doubt it is that this would imply that any of the combinations in your table that produce a 1 would imply that the vertex is not the induced subgraph color. But you only list 4 of the 7 as being able to make a coloring deduction. A similar situation occurs for tri-cycles multiplying to give you 2, where you would be able to deduce instead that the exceptional vertex is the induced subgraph color. Again, there are 6 such instances in the table, but you only point out 3 specific cases as causing deductions. I suppose what makes me most skeptical of my interpretation is the clearly nonsensical situation that occurs when the tri-cycles multiply to 3. For then you should simultaneously be able to deduce that the vertex is and is not the induced subgraph color.

To put my question a different way, when you write:

gsf wrote:This next figure shows the seven tri-cycles that provide coloring information.

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`(0) (0) (0)`

/ \ / \ / \

1 1 1 1 3 3

/ \ / \ / \

*---2---* *---3---* *---2---*

(1) (1) (1) (1)

/ \ / \ / \ / \

2 2 2 3 3 3 2 2

/ \ / \ / \ / \

*---1---* *---1---* *---1---* *---3---*

(0) means that the vertex cannot be the induced subgraph color, and (1) means that the vertex must be the induced subgraph color.

why are these "the" seven tri-cycles...? What doesn't work, for instance, with

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`(0)`

/ \

1 3

/ \

*---2---*

My second question has to do with your example:

gsf wrote:......5.9..7..462.....2..7......6.134.6...8.587.1......2..7.....358..2..1.8......

contains a few y-cycles and y-knots:

y-cycle 4 5 a/9 [96]-[66][65][25]=[98] => [96]-[78][98]-

denotes a Y cycle on color 4 of size 5 (5 vertices, 5 edges). 1-edges are denoted by "-" ([96]-[66]), 2-edges are denoted by "=" ([25]=[98]), and 3-edges are denoted by adjacent vertex labels ([66][65]). This cycle collapses to a tri-cycle with two 1-edges ([96]-[78] and [98]-[96]) and one 3-edge ([78][98]). From the tri-cycle figure above vertex (cell) [96] cannot be the color 4.

What is "color 4"? Looking at my pencilmark grid, it doesn't appear to be the number 4.

Have you already suppressed the links giving the 2-edge between [25] and [98] or is there a brutally obvious reason for them being 2-linked?

Using your multiplication table, I would view the above deduction as (133)21 =>121 implying the exceptional vertex, [96], is not color 4. Where does the tri-cycle [96]-[78][98]- come from? [78] isn't even in the original cycle.

Hmm. Maybe I will stop now and let you respond.