Horizontal graph (and more)

Every digit d of the sequence S is at the vertex of an infinite graph that visits all integers of S.

This digit d is linked by d edges to the graph.

S is the lexicographically earliest sequence with this property.

I hope S starts like that:

S = 1, 2, 3, 4, 5, 6, 7, 8, 9, 62, 22, 23, 32, 24, 42, 25, 52, 26, 63, 33, 34, 43, 35, 53, 36, 64, 44, 45, 54, 46, 65, 55, 56, 66, 67, 72, 27, 73, 37, 74, 47, ...

The hereunder image will show what we mean.

The first line alternates the blue and red colors for the integers of S

The second line splits the digits of the integers > 9

The third line shows the number of edges involving each digit d (those are the small numbers between the split digits)

Every “big” digit d is surrounded by two “small” digits whose sum is d – this sum being the total number of edges involving d:

 

I don't know if the metaphor of the graph is appropriate for the mechanism ruling S. Neither do I know whether the words “vertex” and “edge” were used correctly in this context. But I think that the last line above explains everything.

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Next morning update

Jean-Marc Falcoz was quick to correct my attempt, extend S (523 terms hereunder) and propose two (fractal?) nice graphs:

S  =  1,2,3,4,5,6,7,8,9,62,22,23,32,24,42,25,52,26,63,33,34,43,35,53,36,64,44,45,54,46,65,55,56,66,67,72,27,73,37,74,47,75,57,76,68,82,28,83,38,84,48,85,58,86,69,92,29,93,39,94,49,95,59,96,77,78,79,97,87,88,89,98,99,622,222,223,322,224,422,225,522,226,623,323,324,423,325,523,326,624,424,425,524,426,625,525,526,626,627,722,227,723,327,724,427,725,527,726,628,822,228,823,328,824,428,825,528,826,629,922,229,923,329,924,429,925,529,926,633,233,234,333,235,433,236,533,237,634,334,335,434,336,534,337,635,435,436,535,437,636,536,537,637,638,727,728,827,729,927,733,238,734,338,735,438,736,538,737,639,828,829,928,833,239,834,339,835,439,836,539,837,644,244,245,344,246,444,247,544,248,645,345,346,445,347,545,348,646,446,447,546,448,647,547,548,648,649,738,739,838,744,249,745,349,746,449,747,549,748,655,255,256,355,257,455,258,555,259,656,356,357,456,358,556,359,657,457,458,557,459,658,558,559,659,666,266,267,366,268,466,269,566,277,278,367,368,467,369,567,377,279,468,469,568,477,288,289,378,379,478,388,299,332,343,354,365,376,387,389,479,488,398,399,432,442,443,454,453,464,465,476,475,486,487,489,499,497,498,542,552,553,543,554,564,563,565,532,569,839,844,574,575,576,577,578,579,588,585,586,587,589,599,596,597,598,652,662,663,653,654,642,664,643,665,667,632,668,845,669,846,675,673,674,676,677,678,679,687,684,685,686,688,689,697,695,696,698,699,752,772,773,762,774,753,763,764,754,755,742,775,743,765,749,929,933,776,732,777,756,766,757,758,759,767,768,769,778,779,787,783,784,785,786,788,789,797,794,795,796,798,799,862,882,883,872,884,863,873,874,864,865,852,885,853,875,854,866,842,886,843,876,847,832,887,848,849,855,888,856,877,857,858,859,867,868,869,878,879,889,895,893,894,896,897,898,899,962,992,993,982,994,972,995,963,983,984,973,985,964,974,975,965,966,952,996,953,986,954,976,955,967,942,997,943,987,944,977,945,968,932,998,934,978,935,969,936,979,937,946,956,957,947,948,938,939,949,958,959,988,989,999,2222,2223,3222,2224,4222,2225,5222,2226,6222,2227,7222,2228,8222,2229,9222,2233,2234,3223,3224,4223,3225,5223,...

1000 terms

10000 terms

And now for something completely different!

(submitted a few minutes ago to the OEIS, here).

> Erasing the digits that are not at the extremities of a(n) does not change the general succession of the sequence's digits, with a(1) = 99.

This is the lexicographically earliest sequence of distinct terms > 98 with this property.

The sequence was started with a(1) = 99 because starting it with a(1) = 1 would have filled the Data section with the first 99 natural numbers.

(...)

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Evening update:

Jean-Marc Falcoz was quick to compute this new seq T and correct my (now erased) attempt! Follow the OEIS link above to see his changes. Many thanks Jean-Marc!

T = 111,121,201,210,112,120,131,211,2010,113,122,141,220,1010,151,311,202,114,132,230,1020,1030,115,123,161,240,221,124,133,212,3010,1040,2020,1050,3020,171,511,203,116,142,410,222,181,204,143,302,152,3030,1060,1070,4010,2030,2040,1080,5010,3040,2050,117,125,191,250,301,126,134,214,1090,232,231,801,260,401,413,3050,241,512,3060,3070,1110,6010,1120,7010,4020,1130,2060,3080,2070,4030,1140,8010,5020,1150,3090,4040,2080,5030,1011,701,215,119,162,5040,3110,1021,206,153,402,144,1160,9010,213,242,321,810,172,610,4050,154,163,3120,5050,224,135,182,310,6020,3130,7020,1031,1170,6030,1180,1041,2090,7030,1190,4060,2110,1051,3140,2120,6040,...

T – 5000 terms