Coloring Locally Bipartite Graphs on Surfaces

In this paper we present classes of topological quantum codes (TQC) derived from the embedding of complete bipartite graphs on the corresponding compact surfaces. Every code achieves minimum distance 3 and its encoding rate is such that k/n → 1 as n → ∞.

Probabilistic generation of quantum contextual sets

A new conjecture about minimal imperfect graphs

Binary contraction of graphs

We characterize triangle-free graphs for which there exists a subset of edges that intersects every chordless cycle in an odd number of edges (TF odd-signable graphs). These graphs arise as building blocks of a decomposition theorem (for cap-free odd-signable graphs) obtained by the same authors. We give a polytime algorithm to test membership in this class. This algorithm is itself based on a decomposition theorem. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 204–220, 2000

Maximum genus and chromatic number of graphs

Strong orientations without even directed circuits

The main results of this paper provide an Efficient Polynomial-Time Approximation Scheme (EPTAS) for approximating the genus (and non-orientable genus) of dense graphs. By dense we mean that \(|E(G)|\ge \alpha \, |V(G)|^2\) for some fixed \(\alpha \gt 0\) . While a constant-factor approximation is trivial for this class of graphs, approximations with factor arbitrarily close to 1 need a sophisticated algorithm and complicated mathematical justification. More precisely, we provide an algorithm that for a given (dense) graph G of order n and given \(\varepsilon \gt 0\) , returns an integer g such that G has an embedding in a surface of genus g , and this is ɛ-close to a minimum genus embedding in the sense that the minimum genus \(\mathsf {g}(G)\) of G satisfies: \(\mathsf {g}(G)\le g\le (1+\varepsilon)\mathsf {g}(G)\) . The running time of the algorithm is \(O(f(\varepsilon)\,n^2)\) , where \(f(\cdot)\) is an explicit function. Next, we extend this algorithm to also output an embedding (rotation system) whose genus is g . This second algorithm is an Efficient Polynomial-time Randomized Approximation Scheme (EPRAS) and runs in time \(O(f_1(\varepsilon)\,n^2)\) . Our algorithms are based on the analysis of minimum genus embeddings of quasirandom graphs. We use a general notion of quasirandom graphs [ 25 ]. We start with a regular partition obtained via an algorithmic version of the Szemerédi Regularity Lemma (due to Frieze and Kannan [ 17 ] and to Fox, Lovász, and Zhao [ 14 , 15 ]). We then partition the input graph into a bounded number of quasirandom subgraphs, which are preselected in such a way that they admit embeddings using as many triangles and quadrangles as faces as possible. Here we provide an ɛ-approximation \(\nu (G)\) for the maximum number of edge-disjoint triangles in G . The value \(\nu (G)\) can be computed by solving a linear program whose size is bounded by certain value \(f_2(\varepsilon)\) depending only on ɛ. After solving the linear program, the genus can be approximated (see Corollary 1.7 ). The proof of this result is long and will be of independent interest in topological graph theory.

Classification of nonorientable regular embeddings of complete bipartite graphs

We investigate the complexity of counting Eulerian tours (#ET) and its variations from two perspectives—the complexity of exact counting and the complexity w.r.t. approximation-preserving reductions (AP-reductions [DGGJ04]). We prove that #ET is #P-complete even for planar 4-regular graphs. A closely related problem is that of counting A-trails (#A-trails) in graphs with rotational embedding schemes (so called maps). Kotzig [Kot68] showed that #A-trails can be computed in polynomial time for 4-regular plane graphs (embedding in the plane is equivalent to giving a rotational embedding scheme). We show that for 4-regular maps the problem is #P-hard. Moreover, we show that from the approximation viewpoint #A-trails in 4-regular maps captures the essence of #ET, that is, we give an AP-reduction from #ET in general graphs to #A-trails in 4-regular maps. The reduction uses a fast mixing result for a card shuffling problem [Wil04].

We investigate the complexity of counting Eulerian tours (#ET) and its variations from two perspectives—the complexity of exact counting and the complexity w.r.t. approximation-preserving reductions (AP-reductions, Dyer et al., Algorithmica 38(3):471–500, 2004). We prove that #ET is #P-complete even for planar 4-regular graphs. A closely related problem is that of counting A-trails (#A-trails) in graphs with rotational embedding schemes (so called maps). Kotzig (Theory of Graphs, Proc. Colloq., Tihany, 1966, pp. 219–230, Academic Press, San Diego, 1968) showed that #A-trails can be computed in polynomial time for 4-regular plane graphs (embedding in the plane is equivalent to giving a rotational embedding scheme). We show that for 4-regular maps the problem is #P-hard. Moreover, we show that from the approximation viewpoint #A-trails in 4-regular maps captures the essence of #ET, that is, we give an AP-reduction from #ET in general graphs to #A-trails in 4-regular maps. The reduction uses a fast mixing result for a card shuffling problem (Wilson, Ann. Appl. Probab. 14(1):274–325, 2004).

A method is described for generating and exploring tangled piecewise-linear embeddings of trivalent graphs under the constraints of point-group symmetry. It is shown that the possible vertex-transitive tangles are either graphs of vertex-transitive polyhedra or bipartite vertex-transitive nonplanar graphs. One tangle is found for 6 vertices, three for 8 vertices (tangled cubes), seven for 10 vertices, and 21 for 12 vertices. Also described are four isogonal embeddings of pairs of cubes and 12 triplets of tangled cubes (16 and 24 vertices, respectively). Vertex 2-transitive embeddings are obtained for tangled trivalent graphs with 6 vertices (two found) and 8 vertices (45 found). Symmetrical tangles of the 10-vertex Petersen graph and the 20-vertex Desargues graph are also described. Extensions to periodic tangles are indicated. These are all interesting and viable targets for molecular synthesis.

A classification of orientably-regular embeddings of complete multipartite graphs

Graph embedding has become the research focus in both academic and industrial communities due to its powerful capabilities. The majority of existing work overwhelmingly learn node embeddings in the context of static, plain or attributed, homogeneous graphs. However, many real-world applications frequently involve bipartite graphs with temporal and attributed interaction edges, named temporal interaction graphs. The temporal interactions usually imply different facets of interest and might even evolve over time, thus putting forward huge challenges in learning effective node representations. In this paper, we propose a novel framework named TigeCMN to learn node representations from a sequence of temporal interactions. Specifically, we devise two coupled memory networks to store and update node embeddings in external matrices explicitly and dynamically, which forms deep matrix representations and could enhance the expressiveness of the node embeddings. We conduct experiments on two real-world datasets and the experimental results empirically demonstrate that TigeCMN can outperform the state-of-the-arts with different gains.

Malware are developed for various types of malicious attacks, e.g., to gain access to a user’s private information or control of the computer system. The identification and classification of malware has been extensively studied in academic societies and many companies. Beyond the traditional research areas in this field, including malware detection, malware propagation analysis, and malware family clustering, this paper focuses on identifying the “author group” of a given malware as a means of effective detection and prevention of further malware threats, along with providing evidence for proper legal action. Our framework consists of a malware-feature bipartite graph construction, malware embedding based on DeepWalk, and classification of the target malware based on the k-nearest neighbors (KNN) classification. However, our KNN classifier often faced ambiguous cases, where it should say “I don’t know” rather than attempting to predict something with a high risk of misclassification. Therefore, our framework allows human experts to intervene in the process of classification for the final decision. We also developed a graphical user interface that provides the points of ambiguity for helping human experts to effectively determine the author group of the target malware. We demonstrated the effectiveness of our human-in-the-loop classification framework via extensive experiments using real-world malware data.