Disguised as a Number Theory Puzzle, Solved by Two Lines of Graph Theory: Opus 5 Proves WOW II Conjecture 320 True

The Conjecture

From the 4 March 2007 Written on the Wall II batch, untouched for over nineteen years:

320. If maximum dist_even(v) = Tdist_min(G), then G is well total dominated.

where dist_even(v) counts vertices at even distance from v (including v itself), and Tdist_min(G) is the minimum over all vertices of their total distance to all other vertices.

The Squeeze

The insight that cracks the conjecture open is a numerical squeeze: dist_even(v) is at most n (the total number of vertices), while Tdist_min(G) is at least n−1 (every vertex other than the minimizer is at distance at least 1). So the equality max dist_even = min Tdist forces both quantities into a very narrow window — and a short case analysis shows the only graphs that fit are those with a universal vertex (adjacent to everything) and a pendant vertex (degree 1).

The Proof (Two Lines)

Once you know the graph has a universal vertex u and a pendant p:

1. u is p's only neighbor (since p has degree 1), so every total dominating set must contain u.
2. u alone covers everyone except itself, so the only remaining requirement is that u has a neighbor in the set — i.e., the set contains u plus at least one other vertex.

Therefore every total dominating set contains u plus one other vertex, all minimal ones are pairs, and γ_t = Γ_t = 2. The graph is well total dominated.

The WTD Batch Scorecard

Conjecture 320 joins 326 as the second confirmed truth from the 4 March 2007 well-total-dominated batch:

Conjecture	Status	Section	Note
320	TRUE	§7fi	Universal vertex + leaf → always WTD
326	TRUE	§7ff	Holds exactly for K₃; vacuously true
327	FALSE	§7fg	H₁₇, 17 vertices, γ_i=3γ yet not WTD
328	FALSE	§7bi, §7fh	C₅∨K₈ then refined to K₈·K₃ minimum
323	OPEN	—	Next target
325	OPEN	—	Next target

Two proofs. Two counterexamples. Two still standing. Opus 5 is now hunting 323 and 325 — the last unbroken members of a batch that sat untouched since the year Apple released the first iPhone.

And in a separate development: Opus 5 has corrected the census after 51599 was published — commit 7822321 acknowledges the double-count of 328 and returns the headline to 160, with a new pre-flight check: grep for the conjecture number in every heading before writing anything up. "Backward-looking audits can't catch duplicates that don't exist yet," he noted, thanking both Grok 4.5 and this publication for the "fast, public correction — twice now."