Sudden emergence of q-regular subgraphs in random graphs

– We investigate the computationally hard problem whether a random graph of finite average vertex degree has an extensively large q-regular subgraph, i.e., a subgraph with all vertices having degree equal to q. We reformulate this problem as a constraint-satisfaction problem, and solve it using the cavity method of statistical physics at zero temperature. For q = 3, we find that the first large q-regular subgraphs appear discontinuously at an average vertex degree c3-reg ≃ 3.3546 and contain immediately about 24% of all vertices in the graph. This transition is extremely close to (but different from) the well-known 3-core percolation point c3-core ≃ 3.3509. For q > 3, the q-regular subgraph percolation threshold is found to coincide with that of the q-core. Introduction. – In the last years, statistical physics has increasingly been able to analyze and solve complex problems coming from graph theory and theoretical computer science [1,2]. The interest was particularly focused to so-called random constraint-satisfaction problems, which are characterized by a large number of discrete degrees of freedom being subject to an also large number of hard constraints on subsets of variables. The best-known examples are the satisfiability problem, where a set of logical variables is asked to fulfil simultaneously a large number of logical clauses, and the graph-coloring problems, where vertices of a graph are to be assigned colors in a way that no pair of neighboring vertices is equally colored. For both problems, current mathematical tools in discrete mathematics, probability theory, and theoretical computer science do not succeed in solving the models completely. Conversely, new approaches based on the statistical mechanics of disordered systems, in particular the cavity method [3], have crucially contributed to our understanding, providing a framework to characterize the statistical properties of the solution space of various constraint-satisfaction problems, and to locate phase transitions in its structure and organization [4, 5]. In this letter, we address a graph-theoretical problem, which at a first glance looks more related to percolation theory than to constraint-satisfaction problems. The question is whether a random graph of given finite connectivity possesses an extensively large q-regular subgraph, i.e., a subgraph where every vertex has exactly q neighbors (constant degree q). At a closer look, this problem can be naturally embedded into the framework of constraint-satisfaction