Efficient total colorings of cubic maps of girth 4 and related topics
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916008384528384 |
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| author | Dejter, Italo J |
| author_facet | Dejter, Italo J |
| contents | Let $2\le k\in\mathbb{Z}$. A total coloring of a$k$-regular simple graph via $k+1$ colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. Focus was set upon graphs of girth $k+1$ with efficient total colorings of finite simple cubic graphs $Γ$ of girth 4 built up from the 3-cube and leading to a conjecture that all of those colorings were obtained by means of four basic operations. In the present work, two more basic operations are found necessary in terms of combinatorial cubic maps $M(Γ)$ of which the graphs $Γ$ are their 1-skeletons. This takes to conjecturing that any simple cubic graph that is toroidally 3-edge-connected (defined in the work) and whose $\ell$-belts have $\ell\equiv 0$ mod 4 has an efficient total coloringAn application of one of the two new basic operations yields total perfect code partitions in $g$-toroidal cubic graphs of girth 4 via semi-total colorings that reduce to total colorings, for every $0<g\in\mathbb{Z}$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_02991 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Efficient total colorings of cubic maps of girth 4 and related topics Dejter, Italo J Combinatorics 05C15, 05C38, , 05C69, 05C70, 94B25 Let $2\le k\in\mathbb{Z}$. A total coloring of a$k$-regular simple graph via $k+1$ colors is an efficient total coloring if each color yields an efficient dominating set, where the efficient domination condition applies to the restriction of each color class to the vertex set. Focus was set upon graphs of girth $k+1$ with efficient total colorings of finite simple cubic graphs $Γ$ of girth 4 built up from the 3-cube and leading to a conjecture that all of those colorings were obtained by means of four basic operations. In the present work, two more basic operations are found necessary in terms of combinatorial cubic maps $M(Γ)$ of which the graphs $Γ$ are their 1-skeletons. This takes to conjecturing that any simple cubic graph that is toroidally 3-edge-connected (defined in the work) and whose $\ell$-belts have $\ell\equiv 0$ mod 4 has an efficient total coloringAn application of one of the two new basic operations yields total perfect code partitions in $g$-toroidal cubic graphs of girth 4 via semi-total colorings that reduce to total colorings, for every $0<g\in\mathbb{Z}$. |
| title | Efficient total colorings of cubic maps of girth 4 and related topics |
| topic | Combinatorics 05C15, 05C38, , 05C69, 05C70, 94B25 |
| url | https://arxiv.org/abs/2604.02991 |