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

TL;DR

This paper introduces the list version of ($p$,1)-total labellings in graphs, explores its properties for specific graph classes, and establishes bounds for certain graphs like stars and outerplanar graphs.

Contribution

It defines the ($p$,1)-total labelling choosability, proposes a conjecture on its upper bound, and provides bounds for stars and outerplanar graphs.

Findings

Bound $C_{p,1}^T(K_{1,n}) extless=n+2p-1$ for star graphs.

Bound $C_{p,1}^T(G) extless= ext{max}\{ ext{degree}+2p-1 ext{, for outerplanar graphs.

Proposes a conjecture on the upper bound of the ($p$,1)-total labelling choosability.

Abstract

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.