Locally Irregular Total Colorings of Graphs

TL;DR

This paper investigates locally irregular total colorings of graphs, proving the conjecture for specific graph classes and establishing bounds based on graph properties like chromatic number, planarity, and graph decompositions.

Contribution

It proves the conjecture that two colors suffice for locally irregular total coloring in certain graph classes and provides bounds related to graph chromatic number and planarity.

Findings

Proved the conjecture for cacti, subcubic, and split graphs.

Established bounds based on chromatic number.

Provided constant bounds for planar and outerplanar graphs.

Abstract

A total graph is an ordered triple $(V_0, V_1, E)$, where $V_0, V_1$ are the sets of empty and full vertices, respectively, $V_0 \cap V_1 = \emptyset$, and the set of edges $E$ is a subset of \(\binom{V_0 \cup V_1}{2}\) $(E\cap(V_0 \cup V_1)=\emptyset)$. A simple graph is a total graph in which all vertices are full. We say that a total graph $G$ is locally irregular if every two adjacent vertices have different total degrees, where by the total degree of a vertex $v$ in $G$ we mean the number of edges in $G$ that contain $v$ plus 1 if $v$ is full, or plus 0 if $v$ is empty. A total coloring of a graph $G$ whose colors induce locally irregular total subgraphs is called locally irregular total coloring, and the minimum number of colors required in such a coloring of $G$ is denoted by ${\rm tlir}(G)$. In 2015, Baudon, Bensmail, Przyby{\l}o, and Wo\'zniak conjectured that ${\rm tlir}(G)\leq 2$ for every graph $G$. In this paper, we prove this conjecture for cacti, subcubic graphs, and split graphs. We also provide a general upper bound for ${\rm tlir}(G)$ depending on the chromatic number of $G$, and a constant upper bound if $G$ is planar or outerplanar. In our proofs, we utilize special decompositions of graphs and the connection between acyclic vertex coloring and locally irregular total coloring.