(2,1)-Total labeling of planar graphs with large maximum degree

The ($d$,1)-total labelling of graphs was introduced by Havet and Yu. In this paper, we prove that, for planar graph $G$ with maximum degree $\Delta\geq12$ and $d=2$, the (2,1)-total labelling number $\lambda_2^T(G)$ is at most $\Delta+2$.

💡 Research Summary

The paper studies (d,1)-total labeling, a generalization of total coloring introduced by Havet and Yu. In a (d,1)-total labeling, vertices and edges receive colors from {0,…,k} such that adjacent vertices receive distinct colors, adjacent edges receive distinct colors, and the absolute difference between the color of a vertex and any incident edge is at least d. The smallest k for which such a labeling exists is denoted λ⁽ᵈ⁾ᵀ(G).

When d = 1 the problem coincides with ordinary total coloring, and Havet and Yu conjectured a universal bound λ⁽ᵈ⁾ᵀ(G) ≤ min{Δ + 2d − 1, 2Δ + d − 1} for any graph G (the (d,1)-total labeling conjecture). For d = 2 this becomes λ₂ᵀ(G) ≤ Δ + 3. Prior work established this bound for special classes (outerplanar graphs, planar graphs with large girth, etc.), but the general case for planar graphs remained open.

The authors prove a stronger result: for every planar graph G with maximum degree Δ ≥ 12, the (2,1)-total labeling number satisfies
 Δ + 1 ≤ λ₂ᵀ(G) ≤ Δ + 2.

The lower bound follows immediately from a known inequality λ⁽ᵈ⁾ᵀ(G) ≥ Δ + d − 1 (Proposition 1.4). The main contribution is the upper bound Δ + 2. To achieve this, the authors work with a minimal counterexample G (minimal with respect to |V| + |E|) that would violate the bound. By minimality, every proper subgraph of G admits a (2,1)-total labeling using a palette of size M + 3, where M ≥ 12 is an arbitrary integer with Δ ≤ M.

A series of structural lemmas (Lemmas 2.1–2.6) are proved, revealing stringent restrictions on G:

• Any edge uv satisfies d(u)+d(v) ≥ M − 1 (Lemma 2.1).
• If the smaller endpoint of uv has degree at most ⌊M/4⌋ + 2, then d(u)+d(v) ≥ M + 2 (Lemma 2.2).
• The graph contains no k‑alternator (a bipartite subgraph with prescribed degree constraints) for 3 ≤ k ≤ ⌊M/4⌋ + 2 (Lemma 2.4).
• For each low‑degree vertex set Xₖ (vertices of degree ≤ k) there exists a bipartite “master” subgraph Mₖ linking each x∈Xₖ to a unique neighbor (its k‑master) of higher degree (Lemma 2.5).
• Specific forbidden configurations are identified (Lemma 2.6): a 4‑vertex cannot be adjacent to a vertex of degree ≤ 7; a face with boundary degrees