Locally Irregular Total Colorings of Graphs
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arXiv
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| Natura: | Preprint |
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2026
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| author | Flaszczyńska, Anna Gorzkowska, Aleksandra Grzelec, Igor Onderko, Alfréd Woźniak, Mariusz |
| author_facet | Flaszczyńska, Anna Gorzkowska, Aleksandra Grzelec, Igor Onderko, Alfréd Woźniak, Mariusz |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_13178 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Locally Irregular Total Colorings of Graphs Flaszczyńska, Anna Gorzkowska, Aleksandra Grzelec, Igor Onderko, Alfréd Woźniak, Mariusz Combinatorics 05c15 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. |
| title | Locally Irregular Total Colorings of Graphs |
| topic | Combinatorics 05c15 |
| url | https://arxiv.org/abs/2603.13178 |