Hexagonal Run-Length Zero Capacity Region—Part I: Analytical Proofs

The zero capacity region for hexagonal
$(d,k)$
run-length constraints is known for many, but not all,
$d$
and
$k$
. The pairs
$(d,k)$
for which it has been unproven whether the capacity is zero or positive consist of: (i)
$k=d+2$
when
$d\ge 2$
; (ii)
$k=d+3$
when
$d \ge 1$
; (iii)
$k=d+4$
when either
$d=4$
or
$d$
is odd and
$d \ge 3$
; and (iv)
$k=d+5$
when
$d=4$
. Here, we prove that the capacity is zero for all of case (i), and for case (ii) whenever
$d \ge 7$
. The method used in this paper is to reduce an infinite search space of valid labelings to a finite set of configurations that we exhaustively examine using backtracking. In Part II of this two-part series, we use automated procedures to prove that the capacity is zero in case (i) when
$2 \le d \le 9$
, in case (ii) when
$3 \le d \le 11$
, and in case (iii) when
$d \in \{ 4,5,7,9 \}$
, and that the capacity is positive in case (ii) when
$d \in \{1,2\}$
, in case (iii) when
$d = 3$
, and in case (iv). Thus, the only remaining unknown cases are now when
$k=d+4$
, for any odd
$d \ge 11$
.