Proof. The first equality follows directly from (1). To see the second equality, first let M be a matching of size ν(G). For each of the |V | − 2|M | vertices v missed by M , add to M an edge covering v. We obtain an edge cover of size |M |+ (|V | − 2|M |) = |V | − |M |. Hence ρ(G) ≤ |V | − ν(G). Second, let F be an edge cover of size ρ(G). For each v ∈ V delete from F , dF (v) − 1 edges incide...

We analyse a newspaper market where media …rms compete for advertising as well as for readership. Firms …rst choose the political position of their newspaper, then set cover prices and advertising tari¤s. We build on the duopoly work in two-sided markets of Gabszewicz, Laussel, and Sonnac (2001, 2002) who show that advertising …nancing can lead to minimum political di¤erentiation. We extend the...

An original parallel algorithm is presented that stems from a suitable physical metaphor by considering each edge of the graph as a cell that is attracted by the corresponding vertices and finally moves towards the one of them that wins the competition.

The linkage between economic activity and geography is obvious: Populations cluster mainly on coasts and rarely on ice sheets. Past studies of the relationships between economic activity and geography have been hampered by limited spatial data on economic activity. The present study introduces data on global economic activity, the G-Econ database, which measures economic activity for all large ...

Every 2-connected simple cubic graph of order n ≥ 6 has a cycle cover with at most dn4 e cycles. This bound is best possible.

A k-uniform, d-regular instance of Exact Cover is a family of m sets Fn,d,k = {Sj ⊆ {1, . . . , n}}, where each subset has size k and each 1 ≤ i ≤ n is contained in d of the Sj . It is satisfiable if there is a subset T ⊆ {1, . . . , n} such that |T ∩ Sj | = 1 for all j. Alternately, we can consider it a d-regular instance of Positive 1-in-k SAT, i.e., a Boolean formula with m clauses and n var...

A graph H is a cover of a graph G if there exists a mapping φ from V (H) onto V (G) such that for every vertex v of G, φ maps the neighbours of v in H bijectively onto the neighbours of φ(v) in G. Negami conjectured in 1987 that a connected graph has a finite planar cover if and only if it embeds in the projective plane. This conjecture is not completely solved yet, but partial results due to A...