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A $(2,1)$-total labelling of a simple graph $G$ is a function $\pi \colon V(G)\cup E(G) \to \{0, \ldots, k\}$ such that: $\pi(u) \neq \pi(v)$ for $uv \in E(G)$; $\pi(uv) \neq \pi(vw)$ for $uv, vw \in E(G)$; and $|\pi(uv)-\pi(u)| \geq 2$ and $|\pi(uv)-\pi(v)| \geq 2$ for $uv \in E(G)$. The $(2,1)$-total number $\lambda_2^t(G)$ of $G$ is the least $k$ for which $G$ admits such a labelling. In 2008, Havet and Yu conjectured that $\lambda_2^t(G)\leq 5$ for every connected graph $G \not\cong K_4$ with $\Delta(G) \leq 3$. We prove that, for near-ladder graphs, $\lambda_2^t(G)=5$, verifying Havet and Yu's Conjecture for this class.

A graceful labelling of a tree T is an injective function f: V (T) → {0, 1, . . . , |E(T)|} such that {|f(u)−f(v)|: uv ∈ E(T)} = {1, 2, . . . , |E(T)|}. A tree T is said to be 0-rotatable if, for any v ∈ V (T), there exists a graceful labelling f of T such that f(v) = 0. In this work, it is proved that the follow- ing families of caterpillars are 0-rotatable: caterpillars with perfect matching; caterpillars obtained by identifying a central vertex of a path Pn with a vertex of K2; caterpillars obtained by identifying one leaf of the star K1,s−1 to a leaf of Pn, with n ≥ 4 and s ≥ ⌈n−1 2 ⌉; caterpillars with diameter five or six; and some families of caterpillars with diameter at least seven. This result reinforces the conjecture that all caterpillars with diameter at least five are 0-rotatable.

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