The (2,1)-total number of near-ladder graphs - Omai, M. M.; Campos, C. N.; Luiz, A. G. (Anais do Encontro de Teoria da Computação (ETC); 2021: Anais do VI Encontro de Teoria da Computação, 2021)
A $(2,1)$-total labelling of a simple graph $G$ is a function $π \colon V(G)\cup E(G) → \{0, …, k\}$ such that: $π(u) ≠ π(v)$ for $uv \in E(G)$; $π(uv) ≠ π(vw)$ for $uv, vw \in E(G)$; and $|π(uv)-π(u)| ≥ 2$ and $|π(uv)-π(v)| ≥ 2$ for $uv \in E(G)$. The $(2,1)$-total number $λ_2^t(G)$ of $G$ is the least $k$ for which $G$ admits such a labelling. In 2008, Havet and Yu conjectured that $λ_2^t(G)≤ 5$ for every connected graph $G \not\cong K_4$ with $Δ(G) ≤ 3$. We prove that, for near-ladder graphs, $λ_2^t(G)=5$, verifying Havet and Yu's Conjecture for this class.