Locally Irregular Total Colorings of 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, Przybyło, and Woźniak 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.

Figures (7)

• Figure 1: Locally irregular coloring of the bow-tie graph $B$ using four colors.
• Figure 2: TLIR 2-colorings of $G$ from the case when $x_0 \neq x_3$ and $x_0x_3 \notin E(G)$.
• Figure 3: TLIR 2-colorings of $G$ from the case when $x_0 \neq x_3$, but $x_0x_3 \in E(G)$ in the first subcase.
• Figure 4: TLIR 2-colorings of $G$ from the case when $x_0 = x_3$.
• Figure 5: A copy of the graph $W$ attached by an edge $xw_0$ to a vertex $x \in X$ from $G$.

Theorems & Definitions (33)

• Conjecture 1: 1-2-3 Conjecture
• Conjecture 2: 1-2 Conjecture
• Conjecture 3: Local Irregularity Conjecture Baudon2015, Sedlar2024
• Conjecture 4
• Proposition 5
• proof
• Proposition 6: Corollary of Theorem 1 in Deng2025
• Lemma 7
• proof
• Theorem 10