Random @math -intersection graphs have recently received much interest in a wide range of application areas. Broadly speaking, a random @math -intersection graph is constructed by first assigning each vertex a set of items in some random manner, and then putting an undirected edge between all pairs of vertices that share at least @math items (the graph is called a random intersection graph when @math ). A special case of particular interest is a uniform random @math -intersection graph, where each vertex independently selects the same number of items uniformly at random from a common item pool. Another important case is a binomial random @math -intersection graph, where each item from a pool is independently assigned to each vertex with the same probability. Both models have found numerous applications thus far including cryptanalysis, and the modeling of recommender systems, secure sensor networks, online social networks, trust networks and small-world networks (uniform random @math -intersection graphs), as well as clustering analysis, classification, and the design of integrated circuits (binomial random @math -intersection graphs). In this paper, for binomial uniform random @math -intersection graphs, we present results related to @math -connectivity and minimum vertex degree. Specifically, we derive the asymptotically exact probabilities and zero-one laws for the following three properties: (i) @math -vertex-connectivity, (ii) @math -edge-connectivity and (iii) the property of minimum vertex degree being at least @math .
For graph @math , Bloznelis and uczak @cite_17 recently consider the following three properties: (i) the graph has a minimum vertex degree at least @math ; (ii) the graph has a perfect matching; (iii) the graph is connected. They present that when @math is even, for each of these three properties, its probability converges to @math as @math , under conditions @math ( @math is the edge probability), @math and @math for some @math .
@cite_17
2020272293
A graph @math is called @math -full if @math contains a subgraph @math of @math edges such that, for any non-empty @math , @math holds. Here, @math denotes the set of vertices incident to @math . It is known that the family of edge sets of @math -full graphs forms a family of matroid, known as the sparsity matroid of @math . In this paper, we give a constant-time approximation algorithm for the rank of the sparsity matroid of a degree-bounded undirected graph. This leads to a constant-time tester for @math -fullness in the bounded-degree model, (i.e., we can decide with high probability whether an input graph satisfies a property @math or far from @math ). Depending on the values of @math and @math , it can test various properties of a graph such as connectivity, rigidity, and how many spanning trees can be packed. Based on this result, we also propose a constant-time tester for @math -edge-connected-orientability in the bounded-degree model, where an undirected graph @math is called @math -edge-connected-orientable if there exists an orientation @math of @math with a vertex @math such that @math contains @math arc-disjoint dipaths from @math to each vertex @math and @math arc-disjoint dipaths from each vertex @math to @math . A tester is called a one-sided error tester for @math if it always accepts a graph satisfying @math . We show, for @math and (proper) @math , any one-sided error tester for @math -fullness and @math -edge-connected-orientability requires @math queries.