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We propose a new proof technique that aims to be applied to the same problems as the Lov\'asz Local Lemma or the entropy-compression method. We present this approach in the context of non-repetitive colorings and we use it to improve…
The problem of spanning trees is closely related to various interesting problems in the area of statistical physics, but determining the number of spanning trees in general networks is computationally intractable. In this paper, we perform…
In recent years, there has been significant interest in characterizing the induced subgraph obstructions to bounded treewidth and pathwidth. While this has recently been resolved for pathwidth, the case of treewidth remains open, and prior…
We demonstrate that graph-based models are fully capable of representing higher-order interactions, and have a long history of being used for precisely this purpose. This stands in contrast to a common claim in the recent literature on…
The observation that some subgraphs, called motifs, appear more often in real networks than in their randomized counterparts has attracted much attention in the scientific community. In the prevalent approach the detection of motifs is…
Degree distribution, or equivalently called degree sequence, has been commonly used to be one of most significant measures for studying a large number of complex networks with which some well-known results have been obtained. By contrast,…
Random Forests (RF) is a popular machine learning method for classification and regression problems. It involves a bagging application to decision tree models. One of the primary advantages of the Random Forests model is the reduction in…
Decision trees have long been recognized as models of choice in sensitive applications where interpretability is of paramount importance. In this paper, we examine the computational ability of Boolean decision trees in deriving, minimizing,…
Let $G$ be an undirected graph. We say that $G$ contains a ladder of length $k$ if the $2 \times (k+1)$ grid graph is an induced subgraph of $G$ that is only connected to the rest of $G$ via its four cornerpoints. We prove that if all the…
It is known that any planar graph with diameter D has treewidth O(D), and this fact has been used as the basis for several planar graph algorithms. We investigate the extent to which similar relations hold in other graph families. We show…
Loops are subgraphs responsible for the multiplicity of paths going from one to another generic node in a given network. In this paper we present an analytic approach for the evaluation of the average number of loops in random scale-free…
This paper considers limit theorems associated with subgraph counts in the age-dependent random connection model. First, we identify regimes where the count of sub-trees converges weakly to a stable random variable under suitable…
Recent large-scale reasoning models have achieved state-of-the-art performance on challenging mathematical benchmarks, yet the internal mechanisms underlying their success remain poorly understood. In this work, we introduce the notion of a…
Small-world graphs, which combine randomized and structured elements, are seen as prevalent in nature. Jon Kleinberg showed that in some graphs of this type it is possible to route, or navigate, between vertices in few steps even with very…
The local tree-width of a graph G=(V,E) is the function ltw^G: N -> N that associates with every natural number r the maximal tree-width of an r-neighborhood in G. Our main graph theoretic result is a decomposition theorem for graphs with…
Models based on preferential attachment have had much success in reproducing the power law degree distributions which seem ubiquitous in both natural and engineered systems. Here, rather than assuming preferential attachment, we give an…
Coloring is a notoriously hard problem, and even more so in the online setting, where each arriving vertex has to be colored immediately and irrevocably. Already on trees, which are trivially two-colorable, it is impossible to achieve…
We study preferential attachment models where vertices enter the network with i.i.d. random numbers of edges that we call the out-degree. We identify the local limit of such models, substantially extending the work of Berger et al.(2014).…
In this paper we introduce a new model of random spanning trees that we call choice spanning trees, constructed from so-called choice random walks. These are random walks for which each step is chosen from a subset of random options,…
Treewidth is a well-known graph invariant with multiple interesting applications in combinatorics. On the practical side, many NP-complete problems are polynomial-time (sometimes even linear-time) solvable on graphs of bounded treewidth. On…