In this paper we study the existence of homomorphisms @math using semidefinite programming. Specifically, we use the vector chromatic number of a graph, defined as the smallest real number @math for which there exists an assignment of unit vectors @math to its vertices such that @math when @math . Our approach allows to reprove, without using the Erdős-Ko-Rado Theorem, that for @math the Kneser graph @math and the @math -Kneser graph @math are cores, and furthermore, that for @math there exists a homomorphism @math if and only if @math divides @math . In terms of new applications, we show that the even-weight component of the distance @math -graph of the @math -cube @math is a core and also, that non-bipartite Taylor graphs are cores. Additionally, we give a necessary and sufficient condition for the existence of homomorphisms @math when @math . Lastly, we show that if a 2-walk-regular graph (which is non-bipartite and not complete multipartite) has a unique optimal vector coloring, it is a core. Based on this sufficient condition we conducted a computational study on Ted Spence's list of strongly regular graphs and found that at least 84 are cores.
For an arbitrary graph @math , the is the vertex minimal subgraph to which @math admits a homomorphism. Every graph has a unique core, and the core of @math is itself a core. Moreover, @math and @math have the same core if and only if they are , i.e., @math and @math . Cores are the unique minimal elements of these homomorphic equivalence classes. In this sense, the core of a graph is the smallest graph retaining all of its homomorphic information. In we show that if @math is UVC, and its unique optimal vector coloring is injective on its vertex neighborhoods, then @math is a core (Theorem ). To illustrate the usefulness of our sufficient condition, we given an easy proof of the fact that for @math , both @math and @math are cores. This is well-known @cite_11 , but our proof avoids invoking the Erd o s-Ko-Rado Theorem to describe the structure of the maximum independent sets of these graphs, and it also avoids using the No Homomorphism Lemma. As a second application, we also show that the graphs @math constructed in are cores.
@cite_11
2070948578 2963040959 2035720072 2964292545
Let @math be a 3-uniform hypergraph of order @math and size @math , and let @math be a subset of vertices of @math . The set @math is a strong transversal in @math if @math contains at least two vertices from every edge of @math . The strong transversal number @math of @math is the minimum size of a strong transversal in @math . We show that @math , and we characterize the hypergraphs that achieve equality in this bound. In particular, we show that the Fano plane is the only connected 3-uniform hypergraph @math of order @math and size @math that achieves equality in this bound. A set @math of vertices in a graph @math is a double total dominating set of @math if every vertex of @math is adjacent to at least two vertices in @math . The minimum cardinality of a double total dominating set of @math is the double total domination number @math of @math . Let @math be a connected graph of order @math with minimum degree at least three. As an application of our hypergraph results, we show that @math with equality if and only if @math is the Heawood graph (equivalently, the incidence bipartite graph of the Fano plane). Further if @math is not the Heawood graph, we show that @math , while if @math is a cubic graph different from the Heawood graph, we show that @math , and this bound is sharp. The k-core of a graph is its maximal subgraph with minimum degree at least k. In this paper, we address robustness questions about k-cores (with fixed k>=3). Given a k-core, remove one edge uniformly at random and find its new k-core. We are interested in how many vertices are deleted from the original k-core to find the new one. This can be seen as a measure of robustness of the original k-core. We prove that, if the initial k-core is chosen uniformly at random from the k-cores with n vertices and m edges, its robustness depends essentially on its average degree c. We prove that, if c->k, then the new k-core is empty with probability 1-o(1). We define a constant c"k^' such that when [email protected]c"k^'[email protected](n) with @j(n)[email protected](n^-^1^ ^2logn), @j(n)>0 and c is bounded, then the probability that the new k-core has less than n-h(n) vertices goes to zero, for any function h(n)-> . The k-core of a graph G is the maximal subgraph of G having minimum degree at least k. In 1996, Pittel, Spencer and Wormald found the threshold λc for the emergence of a non-trivial k-core in the random graph G(n, λ n), and the asymptotic size of the k-core above the threshold. We give a new proof of this result using a local coupling of the graph to a suitable branching process. This proof extends to a general model of inhomogeneous random graphs with independence between the edges. As an example, we study the k-core in a certain power-law or 'scale-free' graph with a parameter c controlling the overall density of edges. For each k ≥ 3, we find the threshold value of c at which the k-core emerges, and the fraction of vertices in the k-core when c is e above the threshold. In contrast to G(n, λ n), this fraction tends to 0 as e➝0. We consider the problem of clustering a graph @math into two communities by observing a subset of the vertex correlations. Specifically, we consider the inverse problem with observed variables @math , where @math is the incidence matrix of a graph @math , @math is the vector of unknown vertex variables (with a uniform prior), and @math is a noise vector with Bernoulli @math i.i.d. entries. All variables and operations are Boolean. This model is motivated by coding, synchronization, and community detection problems. In particular, it corresponds to a stochastic block model or a correlation clustering problem with two communities and censored edges. Without noise, exact recovery (up to global flip) of @math is possible if and only the graph @math is connected, with a sharp threshold at the edge probability @math for Erdős-Renyi random graphs. The first goal of this paper is to determine how the edge probability @math needs to scale to allow exact recovery in the presence of noise. Defining the degree rate of the graph by @math , it is shown that exact recovery is possible if and only if @math . In other words, @math is the information theoretic threshold for exact recovery at low-SNR. In addition, an efficient recovery algorithm based on semidefinite programming is proposed and shown to succeed in the threshold regime up to twice the optimal rate. For a deterministic graph @math , defining the degree rate as @math , where @math is the minimum degree of the graph, it is shown that the proposed method achieves the rate @math , where @math is the spectral gap of the graph @math .