On packing total coloring

TL;DR

This paper introduces the concept of packing total coloring in graphs, extending packing coloring to vertices and edges, and provides bounds and characterizations related to the packing total chromatic number.

Contribution

It defines the new packing total coloring concept, establishes bounds, and characterizes graphs with small packing total chromatic numbers.

Findings

Defined the packing total chromatic number for graphs.

Provided bounds for the packing total chromatic number.

Characterized graphs with packing total chromatic numbers 1 to 5.

Abstract

In this paper, we introduce a new concept in graph coloring, namely the \textit{packing total coloring}, which extends the idea of packing coloring to both the vertices and the edges of a given graph. More precisely, for a graph $G$, a packing total coloring is a mapping $c: V(G) \cup E(G) \rightarrow \{1, 2, \ldots\}$ with the property that for any integer $i$, any two distinct elements $A, B \in V(G) \cup E(G)$ with $c(A) = c(B) = i$ must be at distance at least $i+1$ from each other. Note that the distance between $A$ and $B$ means: a) the usual shortest-path distance between $A$ and $B$ if $A, B \in V(G)$; b) the $\min \{d(a,d), d(a,c),d(b,c), d(b,d)\}+1$ if $\{A, B\} =\{ab, cd\} \subseteq E(G)$; c) the $ \min \{d(a,X), d(b,X)\}+1$ if $\{A, B\}=\{ab, X\}$, where $ab \in E(G)$ and $X \in V(G)$. The smallest integer $k$ such that $G$ admits a packing total coloring using $k$ colors is called the \textit{packing total chromatic number}, denoted by $\chi_\rho^{''}(G)$. In addition to introducing this new concept, we provide lower and upper bounds for the packing total chromatic numbers of graphs. Furthermore, we consider packing total chromatic numbers of graphs from the perspective of their maximum degrees and characterize all graphs $G$ with $\chi_\rho^{''}(G) \in \{1, 2, 3, 4, 5\}$.