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DAM

2010

2010

An equivalence graph is a disjoint union of cliques, and the equivalence number eq(G) of a graph G is the minimum number of equivalence subgraphs needed to cover the edges of G. We consider the equivalence number of a line graph, giving improved upper and lower bounds: 1 3 log2 log2 (G) < eq(L(G)) 2 log2 log2 (G) + 2. This disproves a recent conjecture that eq(L(G)) is at most three for trianglefree G; indeed it can be arbitrarily large. To bound eq(L(G)) we bound the closely-related invariant (G), which is the minimum number of orientations of G such that for any two edges e, f incident to some vertex v, both e and f are oriented out of v in some orientation. When G is triangle-free, (G) = eq(L(G)). We prove that even when G is triangle-free, it is NP-complete to decide whether or not (G) 3.

Related Content

Added |
10 Dec 2010 |

Updated |
10 Dec 2010 |

Type |
Journal |

Year |
2010 |

Where |
DAM |

Authors |
Louis Esperet, John Gimbel, Andrew King |

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