List version of ($p$,1)-total labellings

The ($p$,1)-total number $\lambda_p^T(G)$ of a graph $G$ is the width of the smallest range of integers that suffices to label the vertices and the edges of $G$ such that no two adjacent vertices have the same label, no two incident edges have the same label and the difference between the labels of a vertex and its incident edges is at least $p$. In this paper we consider the list version. Let $L(x)$ be a list of possible colors for all $x\in V(G)\cup E(G)$. Define $C_{p,1}^T(G)$ to be the smallest integer $k$ such that for every list assignment with $|L(x)|=k$ for all $x\in V(G)\cup E(G)$, $G$ has a ($p$,1)-total labelling $c$ such that $c(x)\in L(x)$ for all $x\in V(G)\cup E(G)$. We call $C_{p,1}^T(G)$ the ($p$,1)-total labelling choosability and $G$ is list $L$-($p$,1)-total labelable. In this paper, we present a conjecture on the upper bound of $C_{p,1}^T$. Furthermore, we study this parameter for paths and trees in Section 2. We also prove that $C_{p,1}^T(K_{1,n})\leq n+2p-1$ for star $K_{1,n}$ with $p\geq2, n\geq3$ in Section 3 and $C_{p,1}^T(G)\leq \Delta+2p-1$ for outerplanar graph with $\Delta\geq p+3$ in Section 4.

💡 Research Summary

The paper extends the concept of (p, 1)-total labelling of graphs to a list‑colouring framework. A (p, 1)-total labelling assigns integers to both vertices and edges such that adjacent vertices receive distinct labels, adjacent edges receive distinct labels, and the absolute difference between a vertex label and any incident edge label is at least p. The classical parameter λ_T^p(G) is the smallest integer k for which a (p, 1)-total labelling using the colour set {0,…,k} exists.

The authors introduce the list version: for each element x∈V(G)∪E(G) a list L(x) of admissible colours is given, all of the same size k. The graph is said to be k‑(p, 1)-total‑choosable if, for every such k‑assignment, a (p, 1)-total labelling respecting the lists exists. The minimum such k is denoted C_T^{p,1}(G) and called the (p, 1)-total labelling choosability. Clearly C_T^{p,1}(G) ≥ χ_{p,1}^ℓ(G) ≥ χ_{p,1}(G), where χ_{p,1}^ℓ(G) is the list version of the L(p,1)-labelling of the incidence graph S_I(G). Observation 1 shows that C_T^{p,1}(G) equals the list L(p,1)-colouring number of S_I(G), linking the problem to the well‑studied L(p,q)‑labelling literature.

Paths and Trees.
Using known results for L(p,1)-labelling of paths (Lemma 2.1) the authors first compute χ_T^{p,1}(P_k)=p+3 for k≥3. A simple greedy algorithm yields C_T^{p,1}(P_k) ≤ 2p+1, and for k>p they prove the tighter bound 2p ≤ C_T^{p,1}(P_k) ≤ 2p+1 (Theorem 2.4). Thus, for long paths the list requirement can be roughly twice the parameter p, which is substantially larger than the ordinary total labelling number when p≥4.

For trees, Lemma 2.5 (a known bound for L(d,s)-list labelling) together with Observation 1 gives C_T^{p,1}(T) ≤ Δ+2p−1 for any tree T (Theorem 2.6). Lemma 2.7 shows that this bound is tight when the tree contains a vertex of maximum degree Δ whose neighbours also have degree Δ, confirming optimality for many trees.

Stars.
The paper improves the general tree bound for stars. Theorem 3.1 proves C_T^{p,1}(K_{1,n}) ≤ n+2p−1 for p≥2 and n≥3. The proof colours the centre vertex with the smallest colour in its list, removes the forbidden p‑neighbourhood from the incident edge lists, and then colours the edges greedily; the remaining vertex‑lists are always non‑empty. Lemma 3.2 (a result on λ_T^p for bipartite graphs) together with Theorem 3.3 determines the ordinary (p, 1)-total chromatic number of a star: χ_T^{p,1}(K_{1,n}) = n+p when p<n and = n+p+1 when p≥n. For p=2 the list bound matches the ordinary bound exactly (C_T^{2,1}(K_{1,n}) = n+2), showing the star bound is tight in that case.

Outerplanar Graphs.
A structural lemma (Lemma 4.2) states that any outerplanar graph with minimum degree 2 contains one of three configurations (two adjacent 2‑vertices, a 3‑face with a degree‑2 vertex, or two 3‑faces sharing a degree‑4 vertex). Using a minimal counterexample argument, the authors show that a minimal non‑choosable outerplanar graph H must have δ(H)≥2, and then treat each configuration separately. By carefully deleting colours from the lists of the involved vertices and edges, they guarantee enough colours remain to extend a partial labelling to the whole graph. Consequently, Theorem 4.1 establishes C_T^{p,1}(G) ≤ Δ+2p−1 for any outerplanar graph with maximum degree Δ ≥ p+3. This mirrors the known bound λ_T^p(G) ≤ Δ+2p−1 for ordinary total labelling, confirming that the list version does not require extra colours for this class.

General Conjecture.
Motivated by the results above, the authors propose Conjecture 2.8: for every simple graph G with maximum degree Δ, C_T^{p,1}(G) ≤ Δ+2p. This is a natural relaxation of Havet and Yu’s conjecture λ_T^p(G) ≤ Δ+2p−1. The conjecture is tight for complete graphs, because λ_T^p(K_n)=n+2p−2 and therefore C_T^{p,1}(K_n) ≥ Δ+2p. No counter‑examples are known; all families studied in the paper satisfy the conjectured bound.

Significance and Outlook.
The work systematically builds a bridge between (p, 1)-total labelling and list colouring, providing concrete upper bounds for several fundamental graph families. The technique of translating the problem to the incidence graph and then applying known L(p,q)-list results is particularly powerful and suggests a pathway for tackling more complex families such as planar graphs of higher girth or graphs with bounded treewidth. Future research directions include proving Conjecture 2.8 in full generality, tightening the outerplanar bound to Δ+2p−1 (removing the Δ≥p+3 restriction), and exploring algorithmic aspects of constructing list‑compatible (p, 1)-total labellings efficiently.