Related papers: Generation and Properties of Snarks

An acyclic edge coloring of a graph $G$ is a proper edge coloring such that no bichromatic cycles are produced. The acyclic edge coloring conjecture by Fiam{\v{c}}ik (1978) and Alon, Sudakov and Zaks (2001) states that every simple graph…

$\mathbf F_2$-linear pseudorandom number generators are very popular due to their high speed, to the ease with which generators with a sizable state space can be created, and to their provable theoretical properties. However, they suffer…

Graph Neural Networks (GNNs) achieve state-of-the-art performance in various graph-related tasks. However, the black-box nature often limits their interpretability and trustworthiness. Numerous explainability methods have been proposed to…

In graph theory, a Snark is a connected, bridgeless, Cubic graph that cannot be edge-colored with only three colors. Additionally, to avoid some trivial cases, a Snark is typically required to have a girth of minimum five and a cyclic…

We consider feature selection for applications in machine learning where the dimensionality of the data is so large that it exceeds the working memory of the (local) computing machine. Unfortunately, current large-scale sketching algorithms…

Graph neural networks (GNNs) are able to leverage the structure of graph data by passing messages along the edges of the graph. While this allows GNNs to learn features depending on the graph structure, for certain graph topologies it leads…

For a graph $F$, a graph $G$ is \emph{$F$-free} if it does not contain an induced subgraph isomorphic to $F$. For two graphs $G$ and $H$, an \emph{$H$-coloring} of $G$ is a mapping $f:V(G)\rightarrow V(H)$ such that for every edge $uv\in…

Generative network models play an important role in algorithm development, scaling studies, network analysis, and realistic system benchmarks for graph data sets. The commonly used graph-based benchmark model R-MAT has some drawbacks…

Graph matching, also known as network alignment, refers to finding a bijection between the vertex sets of two given graphs so as to maximally align their edges. This fundamental computational problem arises frequently in multiple fields…

Random graphs (or networks) have gained a significant increase of interest due to its popularity in modeling and simulating many complex real-world systems. Degree sequence is one of the most important aspects of these systems. Random…

Let $G$ be a bridgeless cubic graph. Consider a list of $k$ 1-factors of $G$. Let $E_i$ be the set of edges contained in precisely $i$ members of the $k$ 1-factors. Let $\mu_k(G)$ be the smallest $|E_0|$ over all lists of $k$ 1-factors of…

A graph $G$ admiting a $2$-factor is \textit{pseudo $2$-factor isomorphic} if the parity of the number of cycles in all its $2$-factors is the same. In [M. Abreu, A.A. Diwan, B. Jackson, D. Labbate and J. Sheehan. Pseudo $2$-factor…

An odd $k$-edge-coloring of a graph $G$ is a (not necessarily proper) edge-coloring with at most $k$ colors such that each non-empty color class induces a graph in which every vertex is of odd degree; similarly, if more than one color per…

A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan,…

Consider a graph with a rotation system, namely, for every vertex, a circular ordering of the incident edges. Given such a graph, an angle cover maps every vertex to a pair of consecutive edges in the ordering -- an angle -- such that each…

Many complex systems involve interactions between more than two agents. Hypergraphs capture these higher-order interactions through hyperedges that may link more than two nodes. We consider the problem of embedding a hypergraph into…

We study "positive" graphs that have a nonnegative homomorphism number into every edge-weighted graph (where the edgeweights may be negative). We conjecture that all positive graphs can be obtained by taking two copies of an arbitrary…

We identify all $[1, \lambda, 8]$-cycle regular $I$-graphs and all $[1, \lambda, 8]$-cycle regular double generalized Petersen graphs. As a consequence we describe linear recognition algorithms for these graph families. Using structural…

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…

Understanding how a vertex relates to a set of vertices is a fundamental task in graph analysis. Given a graph $G$ and a vertex set $X \subseteq V(G)$, consider the collection of subsets of the form $N(u) \cap X$ where $u$ ranges over all…