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Lower bounds for sense of direction in regular graphs
Authors:Paolo?Boldi,Sebastiano?Vigna  author-information"  >  author-information__contact u-icon-before"  >  mailto:vigna@dsi.unimi.it"   title="  vigna@dsi.unimi.it"   itemprop="  email"   data-track="  click"   data-track-action="  Email author"   data-track-label="  "  >Email author
Affiliation:(1) Dipartimento di Scienze dell"rsquo"Informazione, via Comelico 39/41, 20135 Milano, Italy
Abstract:A graph G with n vertices and maximum degree $Delta_G$ cannot be given weak sense of direction using less than $Delta_G$ colours. It is known that n colours are always sufficient, and it was conjectured that just $Delta_G + 1$ are really needed, that is, one more colour is sufficient. Nonetheless, it has been shown [3] that for sufficiently large n there are graphs requiring $Omega(nloglog n/log n)$ more colours than $Delta_G$. In this paper, using recent results in asymptotic graph enumeration, we show that (surprisingly) the same bound holds for regular graphs. We also show that $Omegaleft(d_Gsqrt{loglog d_G}right)$ colours are necessary, where d G is the degree of G.Received: April 2002, Accepted: April 2003, Sebastiano Vigna: Partially supported by the Italian MURST (Finanziamento di iniziative di ricerca ldquodiffusardquo condotte da parte di giovani ricercatori).The results of this paper appeared in a preliminary form in Distributed Computing. 14th International Conference, DISC 2000, Springer-Verlag, 2000.
Keywords:Distributed computing  sense of direction  random graphs  regular graphs
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