On cover graphs and dependent arcs in acyclic orientations

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The present paper deals with two graph parameters related to cover graphs and acyclic orientations of graphs. The parameter c(G) of a graph G, introduced by B. Bollobás, G. Brightwell and J. Nešetřil [Order 3 245-255], is defined as the minimum number of edges one needs to delete from G in order to obtain a cover graph. Extending their results, we prove that, for δ > 0, (1 - δ) 1/1 n2p/2 ≤ c(script G sign n,p ≤ (1 + δ) 1/1 n2p/2 asymptotically almost surely as long as Cn-1+1/1 ≤ p(n) ≤ cn-1+1/1-1 for some positive constants c and C. Here, as usual, script G signn,p is the random graph. Given an acyclic orientation of a graph G, an arc is called dependent if its reversal creates an oriented cycle. Let dmin(G) be the minimum number of dependent arcs in any acyclic orientation of G. We determine the supremum, denoted by rχ,g, of d min(G)/e(G) in the class of graphs G with chromatic number χ and girth g. Namely, we show that rχ,g = (2χ-g+2)/ (2χ). This extends results of D. C. Fisher, K. Fraughnaugh, L. Langley and D. B. West [J. Combin. Theory Ser. B 71 73-78]. © 2005 Cambridge University Press.

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Combinatorics Probability and Computing