2014/09/30 by Élie de Panafieu, de Panafieu, Élie, Vlady Ravelomanana +1
Mathematics · #Combinatorics (math.CO) #FOS: Mathematics #math.CO
paper · pdf · doi:10.48550/arxiv.1409.8424
15 pages, long version of the abstract accepted in Eurocomb 2013
arxiv created 2014/09/30 · arxiv updated 2014/10/01
We introduce a new model of random multigraphs with colored vertices and weighted edges. It is similar to the "inhomogeneous random graph model" of Söderberg (2002), extended by Bollobás, Janson and Riordan (2007). By means of analytic combinatorics, we then analyze the birth of "complex components", which are components with at least two cycles. We apply those results to give a complete picture of the finite size scaling and the critical exponents associated to a rather broad family of decision problems. As applications, we derive new proofs of known results on the 2-colorability problem, already investigated by Pittel and Yeum (2010), and on the enumeration of properly q-colored multigraphs, analyzed by Wright (1972). We also obtain new results on the phase transition of the satisfiability of quantified 2-Xor-formulas, a problem introduced by Creignou, Daudé and Egly (2007).