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**Looseness ranges of triangulations on closed surfaces.**
*(English)*
Zbl 1084.05023

A triangulation of a closed surface is tight if every 3-coloring of its vertices that uses all three colors has a face that is incident with vertices of all three colors. A tight triangulation has its underlying graph complete, but the converse is not true. The looseness of a triangulation is the minimum \(k\) such that every \((k+3)\)–coloring of its vertices using all colors has a face incident with vertices of all three colors. Given a graph \(G\) and a closed surface \(F^2\), define \(\zeta_{\min}\) (\(\zeta_{\max}\)) as the minimum (respectively maximum) looseness over all triangulations \(G\) of \(F^2\).

The authors show that \(\zeta_{\max} - \zeta_{\min} \leq 2 \lfloor (2 - \chi(F^2))/2 \rfloor\), where \(\chi(F^2)\) is the Euler characteristic of \(F^2\). As a corollary, any two triangulations of the projective plane with the same graph have the same looseness.

The authors show that \(\zeta_{\max} - \zeta_{\min} \leq 2 \lfloor (2 - \chi(F^2))/2 \rfloor\), where \(\chi(F^2)\) is the Euler characteristic of \(F^2\). As a corollary, any two triangulations of the projective plane with the same graph have the same looseness.

Reviewer: Dan S. Archdeacon (Burlington)

### MSC:

05C10 | Planar graphs; geometric and topological aspects of graph theory |

### Keywords:

tight
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\textit{S. Negami}, Discrete Math. 303, No. 1--3, 167--174 (2005; Zbl 1084.05023)

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### References:

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