On 0-Rotatable Graceful Caterpillars

An injection f : V ( T ) → { 0 , … , | E ( T ) | } of a tree T is a graceful labelling if { | f ( u ) - f ( v ) | : u v ∈ E ( T ) } = { 1 , … , | E ( T ) | } . Tree T is 0-rotatable if, for any v ∈ V ( T ) , there exists a graceful labelling f of T such that f ( v ) = 0 . In this work, the following families of caterpillars are proved to be 0-rotatable: caterpillars with a perfect matching; caterpillars obtained by linking one leaf of the star K 1 , s - 1 to a leaf of a path P n with n ≥ 3 and s ≥ ⌈ n 2 ⌉ ; caterpillars with diameter five or six; and caterpillars T with diam ( T ) ≥ 7 such that, for every non-leaf vertex v ∈ V ( T ) , the number of leaves adjacent to v is even and is at least 2 + 2 ( ( diam ( T ) - 1 ) mod 2 ) . These results reinforce the conjecture that all caterpillars with diameter at least five are 0-rotatable.