Looseness and Independence Number of Triangulations on Closed Surfaces

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The looseness of a triangulation G on a closed surface F2, denoted by ξ (G), is defined as the minimum number k such that for any surjection c : V (G) → {1, 2, . . . , k + 3}, there is a face uvw of G with c(u), c(v) and c(w) all distinct. We shall bound ξ (G) for triangulations G on closed surfaces by the independence number of G denoted by α(G). In particular, for a triangulation G on the sphere, we have

and this bound is sharp. For a triangulation G on a non-spherical surface F2, we have

ξ (G) ≤ 2α(G) + l(F2) − 2,

where l(F2) = [(2 − χ(F2))/2] with Euler characteristic χ(F2).

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Target audience:

researchers in the fields of: colourings, partitions (general colourings), hereditary properties, independence and dominating structures (sets, paths, cycles, etc.), cycles, local properties, products of graphs

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