The proper hat guessing number of a graph is the largest number of hat colors for which its vertices
have a guaranteed winning guessing strategy when the adversary is restricted to proper
vertex colorings. Each vertex sees only its neighbors'
colors and guesses its own color using a rule fixed in advance. The guesses are simultaneous,
and at least one must be correct for every allowed coloring (Adriaensen et al.
2026).
For the complete graph the value is , and for every tree with at least
three vertices it is 4. Restricting the adversary
to proper colorings distinguishes this invariant from the ordinary hat guessing number.
Protti (2026a, 2026b, 2026c, 2026d) reported values 8, 10, 12, and 14 for with , 6, 7, and 8, respectively. These attain the previously
known upper bound . The strategies were developed with GPT-5.6 Pro and accompanied
by exact finite checks. The releases describe additional adversarial artifact reviews,
but conventional peer review and proof assistant verification had not been reported
as of Sep. 7, 2026. These cases do not resolve the entire family.
is the largest number 
of hat colors for which its vertices
have a guaranteed winning guessing strategy when the adversary is restricted to proper
vertex colorings. Each vertex sees only its neighbors'
colors and guesses its own color using a rule fixed in advance. The guesses are simultaneous,
and at least one must be correct for every allowed coloring (Adriaensen et al.
2026).
the value is 
, and for every tree with at least
three vertices it is 4. Restricting the adversary
to proper colorings distinguishes this invariant from the ordinary hat guessing number.
with 
, 6, 7, and 8, respectively. These attain the previously
known upper bound 
. The strategies were developed with GPT-5.6 Pro and accompanied
by exact finite checks. The releases describe additional adversarial artifact reviews,
but conventional peer review and proof assistant verification had not been reported
as of Sep. 7, 2026. These cases do not resolve the entire family.
." 2026a. 
." 2026b. 
." 2026c. 
." 2026d. 
." 2026.